13th algebraic hyperstructures and its applications...

13th Algebraic Hyperstructures and its Applications (AHA2017)

24-27 July, 2017, Istanbul-Turkey

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FOREWORD

13th Algebraic Hyperstructures and its Applications Conference (AHA2017),

organized by the International Algebraic Hyperstructures Association will take place from

24th July to the 27th July 2017 in Istanbul, a fascinating city built on two Continents, divided

by the Bosphorus Strait and this is one of the greatest cities in the world where you can see a

modern western city combined with a traditional eastern city, it’s a melting pot of many

civilizations and different people.

The series of International Conferences on Algebraic Hyperstructures and

Applications (AHA) aims at bringing together researchers and academics for the presentation

and discussion of novel theories and applications of Algebraic. The conference covers a broad

spectrum of topics related to Algebraic Hyperstructures.

AHA2017 provides an ideal academic platform for researchers and scientists to

present the latest research findings in mathematics. The conference aims to bring together

leading academic scientists, researchers and research scholars to exchange and share their

experiences and research results about mathematics and engineering studies.

We would like to thank to Yildiz Technical University for their invaluable supports.

We would also like to thank to all contributors to conference, especially to keynote speakers

who share their significant scientific knowledge with us, to organizing and scientific

committee for their great effort on evaluating the manuscripts. We do believe and hope that

each contributor will get benefit from the conference.

We hope to see you in 14th Algebraic Hyperstructures and its Applications

Conference (AHA2020) at Romania.

Yours Sincerely,

Prof. Dr. Bayram Ali ERSOY

Chair of AHA2017



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Previous AHA Events

1978, held in Taormina, Italy (November 25-28) Organized by P.Corsini

1983, held in Taormina, Italy (October 21-24) Organized by P.Corsini

1985, held in Udine, Italy (October, 15-18) Organized by P.Corsini

1990, held in Xanthi, Greece (June 27-30) Organized by T.Vougiouklis

1993, held in Iasi, Romania (July 4-10) Organized by M.Stefanescu

1996, held in Prague, Czech Republic (September 1-7) Organized by T.Kepca

1999, held in Taormina, Italy (June 13-19) Organized by R.Migliorato

2002, held in Samothrace, Greece (September 1-9) Organized by T.Vougiouklis

2005, held in Babolsar, Iran (September 1-7) Organized by R.Ameri

2008, held in Brno, South Moravia, Czech Republic (September 3-9) Organized by

S.Hoskova

2011, held in Pescara, Italy (October, 16-21) Organized by A. Maturo

2014, held in Xanthi, Greece (September, 2-7) Organized by S. Spartalis



iv

Conference Organization

Chair

Bayram Ali ERSOY (Turkey)

A. Göksel AĞARGÜN (Turkey)

Scientific Committee

· Piergiulio Corsini (Italy)

· Ivo Rosenbeg (Canada)

· Thomas Vougiouklis (Greece)

· Stephen Comer (U.S.A.)

· Violeta Leoreanu Fotea (Romania)

· Mohhamad Mahdi Zahedi (Iran)

· James Jantosciak (U.S.A)

· Reza Ameri (Iran)

· Stefanos Spartalis (Greece)

· Mirela Stefanescu (Romania)

· Tomas Kepka (Czech Republic)

· Renato Migliorato (Italy)

· Bijan Davvaz (Iran)

· Jan Chvalina (Czech Republic)

· Sarka Hoskova (Czech Republic)

· Antonio Maturo (Italy)

· Nemec Petr (Czech Republic)

· Maria Konstantinidou – Serafimidou

(Greece)

· R.A. Borzooei (Iran)

· Mario De Salvo (Italy)

· Christos Massouros (Greece)

· Mashhoor Refai (Jordan)

· Irina Cristea (Slovenia)

· Demetrious Stratigopoulos (Greece)

· Maria Scafati Tallini (Italy)

· A. R. Ashrafi, (Iran)

· Yuming Feng, (China)

· Rosaria Rota (Italy)

· Achilles Deamalides (Greece)

· Ian Tofan (Romania)

· Yupaporn Kemprasit (Thailand)

· Rita Procesi (Italy)

· Mohammad Reza Darafsheh (Iran)

· Aldo Ventre (Italy)

· Huakang Yang (China)

· Krassimir Atanassov (Bulgaria)

· Bayram Ali Ersoy (Turkey)

· K. P. Shum (China)

· Bal Kishan Dass (India)

· Tariq Mahmood (Pakistan)

· Antonios Kalampakas (Greece)

· Murat Sarı (Turkey)

· Nikolaos Antampoufis (Greece)



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Conference organization Co-chair

Serkan ONAR (Turkey)

Local Organizing Committee

E. Mehmet Özkan (Turkey)

Adem Cengiz Çevikel (Turkey)

Murat Kirişci (Turkey)

İbrahim Demir (Turkey)

Filiz Kanbay (Turkey)

Ünsal Tekir (Turkey)

Mutlu Akar (Turkey)

Pınar Albayrak (Turkey)

Elif Demir (Turkey)

S. Ebru Daş (Turkey)

Murat Turhan (Turkey)

Fatma Çeliker (Turkey)

Ashraf Ahmed (Palestine)

Elif Segah Öztaş (Turkey)

M.Emin Köroğlu (Turkey)

Deniz Sönmez (Turkey)

Seda Akbıyık (Turkey)

Mücahit Akbıyık (Turkey)

Rabia Nagehan Üregen (Turkey)

Elif Aktaş (Turkey)

Melis Bolat (Turkey)

Sanem Yavuz (Turkey)

Emre Ersoy (Turkey)



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Contents

Foreword……………………………………………………………………………………….ii

Previous AHA Events……………………………………………………………………… ...iii

Conference Organization……………………………………………………………………...iv

Bijan Davvaz Some Applications of Algebraic Hyperstructures………………….1

Thomas Vougiouklis The Hv-matrix Representations…………………………………….4

Reza Ameri Some categorical aspects of algebraic hyperstructures……………..6

Piergiulio Corsini Hyperstructures and some of the most recent applications…………8

M. Mehdi Zahedi Weak Closure operations on ideals of a BCK-algebra……………...9

Lumnije Shehu Extensions of Polygroups by Polygroups via Factor Polygroups….10

Murat Alp On crossed polysquares and fundamental relations………………..12

Michal Novák Recent advances in EL-hyperstructures……………………….......15

Sinem Tarsuslu HH∗− Intuitionistic Heyting Valued Ω-Algebra…………………..16

Sinem Tarsuslu Algebraic Approach to Multiplicative Set…………………………17

Najmeh Jafarzadeh On the relation between categories of (m, n)-ary hypermodules

and (m, n)-ary modules……………………………………………19

Yıldıray Çelik Soft Bi-Ideals of Soft LA-Semigroups…………………………….23

Jan Chvalina Sequences of groups and hypergroups of linear ordinary

differential operators……………………………………………….24

Ümit Deniz On Different Approach of Fuzzy Ring Homomorphisms………....25

Mahmood Bakhshi L-hyperstructures………………………………………………….26

M. Golmohamadian New connections between hyperstructures and Graph Theory……31

Akbar Paad Ideals in HvMV-algebras………………………………………….33

Hashem Bordbar Overview on the Height of a Hyperideal in Krasner Hyperrings….37

Hashem Bordbar Theory of Double-framed soft set theory on Hyper BCK-algebra...39

Ioanna Iliou On P-hopes and P-Hv-structures on the plane……………………..40

Banu Pazar Varol On Neutrosophic Linear Spaces…………………………………...41

Nesibe Kesicioğlu The relationships between the orders induced by implications

and uninorms……………………………………………………...42

Nesibe Kesicioğlu A survey on order-equivalent uninorms…………………………...43



vii

Olga Cerbu The reflector functor and lattice L (R)…………………………….44

Deniz Sönmez A note on 2-absorbing δ-primary fuzzy ideals of commutative

Rings……………………………………………………………... 46

Rajab Ali Borzooei Relation Between Hyper EQ-algebras and Some Other

Hyper Structures…………………………………………………..47

Şerife Yılmaz Fuzzy hyperideals in ordered semihyperrings……………………..48

Şerife Yılmaz Fuzzy interior hyperideals in ordered semihyperrings…………….49

Ali Taghavi Hyperhilbert Spaces……………………………………………….50

Dilek Bayrak The Lattice Structure of Subhypergroups of a Hypergroup……….51

Tuğba Arkan Intuitionistic Fuzzy Weakly Prime Ideals…………………………52

Karim Ghadimi Some Results on Tensor Product of Krasner Hypervector Spaces...53

Jafar Azami Fuzzy coprimary submodules and their representation……………57

Güzide Şenel Constructing Topological Hyperspace with Soft Sets……………..58

Gülşah Yeşilkurt Fuzzy Weakly Prime Γ-ideals……………………………………...59

Sanem Yavuz Intuitionistic Fuzzy 2-absorbing Ideals of Commutative Rings…...60

Adem C. Cevikel Transition from Two-Person Zero-Sum Games to Cooperative

Games with Fuzzy Payoffs………………………………………..61

Karim Abbasi On computation of fundamental group of a finite hypergroup…….62

Hossein Shojaei Various kinds of quotient of a canonical hypergroup……………...64

Didem S. Uzay On multipliers of hyper BCC-algebras…………………………….66

Şule Ayar Özbal Derivations on hyperlattices……………………………………….67

Naser Zamani On fuzzy φ-prime ideals…………………………………………...68

Thawhat Changphas On pure hyperideals in ordered semihypergroups…………………69

Afagh Rezazadeh Relation Between Hyper K-algebras and Superlattice

(Hypersemilattice)………………………………………………….70

Tahere Nozari Vague Soft Hypermodules………………………………………....75

Zahra Soltani An introduction to Zero-Divisor Graph of a Commutative

Multiplicative Hyperring…………………………………………..76

Niovi Kehayopulu Some ordered hypersemigroups which enter their properties into

their σ-classes……………………………………………………..77

Serkan Onar Study of Γ-hyperrings by fuzzy hyperideals with respect to a

t-norm……………………………………………………………..78

Kostaq Hila On an algebra of fuzzy m-ary semihypergroups…………………..81



viii

Habib Harizavi On Annihilator in Pseudo BCI-algebras…………………………...83

Krisanthi Naka The embedding of an ordered semihypergroup in terms of

fuzzy sets………………………………………………………….84

Ashraf Abumghaiseeb On δ-Primary Hyperideals of Commutative Semihyperrings……..86

Mutlu Akar On the Vahlen Matrices…………………………………………...87

Abderrahmane Bouchair On the Baireness of function spaces………………………………88

Farida Belhannache Global asymptotic stability of a higher order difference equation..89

Kelaiaia Smail Weak alpha favorability of C(X) with a set open topology………90

Nemat Abazari Curves on Lightlike Cone in Minkowski Space…………………..91

Mehmet E. Köroğlu A class of LCD codes from group rings…………………………..92

Sümeyra Uçar New Blocking Cryptography Models……………………………..93

Nihal Taş A New Coding Theory with Generalized Pell (p,i) – Numbers…..94

Sezer Sorgun Some Problems in Spectral Graph Theory………………………..95

Hakan Küçük On trees which have exactly 4 non-zero Randi¢ eigenvalues…….96

Nurten Bayrak Gürses One-Parameter Planar Motions in Generalized Complex Number

Plane CJ…………………………………………………………...97

Esma Demir Çetin A New Approach to Motions and Surfaces with Zero Curvatures in

Lorentz 3-Space…………………………………………………100

Hülya Aytimur Chen-Ricci and Wintgen Inequalities for Statistical Submanifolds of

Quasi-Constant Curvature……………………………………….101

Mücahit Akbıyık One-Parameter Homothetic Motion on the Galilean Plane……...102

Murat Sarı Comparison of encryption and decryption algorithms through

various approaches………………………………………………103

Çağla Ramis Surfaces with Constant Slope and Tubular Surfaces…………….104

Ufuk Çelik New Contributions to Fixed-Circle Results on S-Metric Spaces..105

Hatice Tozak On The Parallel Ruled Surfaces With B-Darboux Frame……….106

Murat Kirişci A ANFIS Perspective for the diagnosis of type II diabetes……..108

Elif Demir Transitive Operator Algebras and Hyperinvariant Subspaces…..109

Süleyman Demir Reformulation of compressible fluid equations in terms of

Biquaternions……………………………………………………110

Erdal Gül On The Trace Formula for a Differential Operator of Second Order

with Unbounded Operator Coefficients…………………………111



ix

Mustafa Bayram Gücen Practical Stability Analyses of Nonlinear Fuzzy Dynamic Systems

of Unperturbed Systems with Initial Time Difference………….112

Sebahat Ebru Das A Numerical Scheme for Solving Nonlinear Fractional Differential

Equations in the Conformable-Derivative Sense………………..113

Fatma Öztürk Çeliker On invariant ideals on locally convex solid Riesz spaces………114

Pınar Albayrak On weakly compact-friendly operators…………………………115

Murat Turhan Hirota type discretization of Clebsch equations………………..116

Fatma Bulut An h-deformation of the superspace R(1|2) via a contraction….117

Mücahit Akbıyık Euler-Savary’s Formula On Dual Plane………………………...119

Ayten Özkan Mistakes and misconceptions regarding to natural numbers on

secondary Mathematics Education……………………………..121

Filiz Kanbay On a Fuzzy Application of the Particulate Matter Estimation….122



1Yazd University, Department of Mathematics, Yazd, Iran

E-mail: [email protected]

Some Applications of Algebraic Hyperstructures

Bijan Davvaz 1

In this study, we describe the applications of algebraic hyperstructures and survey

related works. Hyperstructures represent a natural extension of algebraic structures and they

were introduced in 1934 by F. Marty. He generalized the notion of groups by defining

hypergroups. Algebraic hyperstructures have many applications in various sciences. In [1],

Corsini and Leoreanu presented some of the numerous applications of algebraic

hyperstructures, especially those from the last fifteen years, to the following subjects:

geometry, hypergraphs, binary relations, lattices, fuzzy sets and rough sets, automata,

cryptography, codes, median algebras, relation algebras, artificial intelligence and

probabilities. The largest class of hyperstructures is the one that satisfies weak axioms, i.e.,

the non-empty intersection replaces the equality. These are called Hv-structures and they were

introduced in 1990 by Vougiouklis [2]. The latter hyperstructures have many applications to

different disciplines like Biology, Chemistry, Physics, and so on. In several papers, Davvaz et

al. [3-9] introduced some chemical examples of hyperstructures. For instance, algebraic

hyperstructures associated to chain reactions; algebraic hyperstructures associated to

dismutation reactions; algebraic hyperstructures associated to redox reactions; hyperstructures

associated to electrochemical cells. Another motivation for the study of hyperstructures comes

from biology. In [10], the main objective of authors is to provide examples of hyperstructures

associated to inheritance. They explored the algebraic hyperstructure that naturally occurs as

genetic information gets passed down through generations. Mathematically, the algebraic

hyperstructures that arise in genetics are very interesting ones. They are generally

commutative and weakly associative. Moreover, many of the algebraic properties of these

hyperstructures have genetic significance. Indeed, there is an interplay between the purely

algebraic hyperstructures and the corresponding genetic properties, that makes the subject so

fascinating. The examples given in [10] indicated that the theory of genetic hyperstructure

algebras is generally worth practicing. Mendel, the father of genetics took the first steps in

defining “contrasting characters, genotypes in F1 and F2 . . . and setting different laws”. The

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genotypes of F2 is dependent on the type of its parents genotype and it follows certain roles.

In [11], the authors analyzed the second generation genotypes of monohybrid and a dihybrid

with a mathematical structure. They used the concept of Hv-semigroup structure in the F2-

genotypes with cross operation and proved that this is an Hv-semigroup. They also

determined the kinds of number of the Hv-subsemigroups of F2-genotypes. In [12], the

authors provided examples about different types of inheritance (Mendelian and Non-

Mendelian inheritance) and relate them to hyperstructures and generalize the work done in

[10]. The feature of hyperstructures allows us to extend this theory into the elementary

particle physics. In [13], the authors have considered one important group of the elementary

particles, Leptons. They have shown this set that along with the interactions between its

members can be described by the algebraic hyperstructure. In [14], Asghari-Larimia and

Davvaz presented a connection between algebraic hyperstructures and number theory. They

introduced a hyperoperation associated to the set of all arithmetic functions and analyzed the

properties of this new hyperoperation. Several characterization theorems are obtained,

especially in connection with multiplicative functions. Then, Al Tahan and Davvaz [15]

constructed a hyperring structure on the set of arithmetic functions.

Keywords: hyperstructure, chemistry, biology, physics, number theory.

2010 AMS Classification: 20N20

References:

1. Corsini, P. and Leoreanu, V., Applications of hyperstructures theory, Advances in

Mathematics, Kluwer Academic Publisher, 2003.

2. Vougiouklis, T., The fundamental relation in hyperrings. The general hyperfield,

Algebraic hyperstructures and applications (Xanthi, 1990), 203-211, World Sci. Publishing,

Teaneck, NJ, 1991.

3. Davvaz, B., Dehghan Nezad, A. and Benvidi, A., Chain reactions as experimental

examples of ternary algebraic hyperstructures, MATCH Communications in Mathematical

and in Computer Chemistry, 65(2), 491-499, 2011.

4. Davvaz, B., Dehghan Nezhad, A. and Benvidi, A., Chemical hyperalgebra: Dismutation

reactions, MATCH Communications in Mathematical and in Computer Chemistry, 67,

55- 63, 2012.

5. Davvaz, B. and Dehghan Nezhad, A., Dismutation reactions as experimental

verifications of ternary algebraic hyperstructures, MATCH Communications in

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Mathematical and in Computer Chemistry, 68, 551-559, 2012.

6. Davvaz, B., Dehghan Nezad, A. and Mazloum-Ardakani, M., Chemical hyperalgebra:

Redox reactions, MATCH Communications in Mathematical and in Computer Chemistry,

71, 323-331, 2014.

7. Davvaz, B., Dehghan Nezad, A., Mazloum-Ardakani, M. and Sheikh-Mohseneib, M.A.,

Describing the algebraic hyperstructure of all elements in radiolytic processes in cement

medium, MATCH Communications in Mathematical and in Computer Chemistry, 72,

375-388, 2014.

8. Davvaz, B., Weak algebraic hyperstructures as a model for interpretation of chemical

reactions, Iranian Journal of Mathematical Chemistry, 7(2), 267-283, 2016.

9. Al Tahan, M. and Davvaz, B., Weak chemical hyperstructures associated to

electrochemical cells, Iranian Journal of Mathematical Chemistry, to appear.

10. Davvaz, B., Dehghan Nezad, A. and Heidari, M.M., Inheritance examples of algebraic

hyperstructures, Information Sciences, 224, 180-187, 2013.

11. Ghadiri, M., Davvaz, B. and Nekouian, R., Hv-Semigroup structure on F2-offspring of a

gene pool, International Journal of Biomathematics, 5(4), 1250011 (13 pages), 2012.

12. Al Tahan, M. and Davvaz, B., Hyperstructures associated to biological inheritance,

Mathematical Biosciences, 285, 112-118, 2017.

13. Dehghan Nezhad, A., Moosavi Nejad, S.M., Nadjafikhah and Davvaz, B., A physical

example of algebraic hyperstructures: Leptons, Indian Journal of Physics, 86(11),

1027-1032, 2012.

14. Asghari-Larimi, M. and Davvaz, B., Hyperstructures associated to arithmetic functions,

ARS Combinatoria, 97, 51-63, 2010.

15. Al Tahan, M. and Davvaz, B., On the existence of hyperrings associated to arithmetic

functions, Journal of Number Theory, 174, 136-149, 2017.

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1 Democritus University of Thrace, School of Education 681 00 Alexandroupolis, Greece


The Hv-matrix Representations

Thomas Vougiouklis1

Emeritus Professor

The Theory of Representations of Hyperstructures was started in mid 80’s but that

time there was not any general definition of hyperfield. The Hv-structures, were introduced in

4th AHA Congress 1990, and at the same time, the general definition of the hyperfield, was

given. Since then the Theory of Representations is refereed mainly on Hv-groups by Hv-

matrices, that is that, the matrices have entries elements of an Hv-field or from an Hv-ring. In

Hv-structures the weak axioms replace the classical axioms of structures by replacing the

‘equality’ by the ‘non empty intersection’. The characteristic property of Hv-structures, is that

a partial order on Hv-structures on the same underline set, is defined. The weak properties

increase extremely the number of hyperstructures defined in the same set, therefore it is

reasonable to find applications in mathematics and in other applied sciences, as well. On the

other side, in order to obtain strict results, we ask from applied sciences to give more axioms

and more restrictions. This is the case, for example, in nuclear physics with Santilli’s iso-

theory. In representation theory the researchers have to treat well almost all the classical

algebraic structures from semigroups to Lie-algebras. We present the problems, some new

results and we give to researchers open problems in mathematics from hyperstructures.

Keywords: Hyperstructures, Hv-structures, Hv-matrix.

2010 AMS Classification: 20N20.

References:

1. Corsini P., Leoreanu V., Application of Hyperstructure Theory, Klower Acad. Publ., 2003.

2. Davvaz B., Leoreanu V., Hyperring Theory and Applications, Int. Academic Press, 2007.

3. Vougiouklis T., The fundamental relation in hyperrings. The general hyperfield, 4thAHA,

Xanthi 1990, World Scientific, (1991), 203-211.

4. Vougiouklis T., Hyperstructures and their Representations, Monographs in Math.,

Hadronic, 1994.

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mailto:[email protected]



1 Democritus University of Thrace, School of Education 681 00 Alexandroupolis, Greece


5. Vougiouklis T., On Hv-rings and Hv-representations, Discrete Math., Elsevier, 208/209,

1999, 615-620.

6. Vougiouklis T., Finite Hv-structures and their representations, Rend. Sem. Mat. Messina

S.II, V.9, 2003, 245-265.

7.Vougiouklis T., Hypermathematics, Hv-structures, hypernumbers, hypermatrices and Lie-

Santilli admissibility, American J. Modern Physics,4(5), 2015, 34-46.

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1School of Mathematics‎, , Department of Mathematics, ‎University of Tehran‎, ‎Tehran‎, ‎Iran.


Some Categorical Aspects of Algebraic Hyperstructures

Reza Ameri1

In this study we briefly discuses on some of algebraic hyperstructures theory in view

point of category theory and present some features of the various hyperstructures such as

hypergroups, hyperrings, hypermodules and etc. In this regards we investigate various

categories of hyperstructures based on various kinds of morphisms, especially on multivalued

homomorphisms. We will proceed by introducing some categorical objects such as, zero

object, product, coproduct and free objects. Finally, we constructs some functors from

categories of hyperstructures to the correspondence classical algebraic category.

Today hyperstructures rapidly developed in view point of theory and application and

many concept of classical algebra are appear in this theory. In parallel to this progress main

questions will rise about and some terminology has been used improper. On the other hands,

some of terminology may be bad used. Also, for study the relationships between

hyperstructures and classical algebra we need to use the exact language to correct

mathematically descriptions of these notions. For example in hyperstructures theory we use

the phrases such as: a hypergroup is a generalization of a group, the class of polygroup is a

generalization of ordinary group. The fundamental relation on a hypergroup is a function

which assign to each hypergroup a group or in general to every hyperalgebra one can assign

an algebra via the fundamental relation. Here naturally give rise some main questions:

What is different between the class of hypergroups and groups?

Are they really mathematically different?

And many other questions which appears to study of algebraic hyperstructures and its

relationship to the related algebraic structures. In this paper we briefly to mention the role of

category theory as a useful tools to answer to these questions as well as we introduce some

categorical objects such as product, kernel, free and etc. in category of Krasner hypermodules.

Keywords: category, hypergroups, hypermodules, fundamental functor, hyperadditive

category.

2010 AMS Classification: 03G99, 06B99, 06F05.

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1School of Mathematics‎, , Department of Mathematics, ‎University of Tehran‎, ‎Tehran‎, ‎Iran.


Acknowledgements:

The author partially has been supported by "Algebraic Hyperstructure Excellence (AHETM),

Tarbiat Modares University, Tehran, Iran" and "Research Center in Algebraic hyperstructures

and Fuzzy Mathematics, University of Mazandaran, Babolsar, Iran".

References:

1. R. Ameri, On the categories of hypergroups and hypermodules, J. Discrete Math. Sci. cryptogr. 6

(2003) 121-132.

2. R. Ameri, M.Norouzi, Prime and primary hyperideales in Krasner (m; n)-hyperring, European J.

Combin. 34(2013)379-390.

3. R. Ameri, M.Norouzi, V. Leoreanu-Fotea, On Prime and primary subhypermodules of (m; n)-

hypermodules, European J. Combin. 44(2015)175-190.

4. SM. Anvariyeh, S. Mirvakili, B. Davvaz, Fundamental relation on (m; n)- hypermodules over (m;

n)-hyperrings, Ars combin. 94(2010)273-288.

5. Z. Belali, SM. Anvariyeh, S. Mirvakili, B. Free and cyclic (m; n)-hypermodules, Tamkang J.

Math.42(2011) 105-118.

6 P. Corsini, Prolegemena of Hypergroup Theory, second ed. Aviani, Editor, 1993.

7. P. Corsini, V. Leoreanu, Applications of Hyperstructure Theory, in: Advances in

Mathematices, Vol. 5, Kluwer Academic Publishers, 2003.

8. G. Crombez, On (m; n)-rings, Abh. Math. Sem. Univ. Hamburg 37(1972)180-199.

9. G. Crombez, J. Timm, On (m;n)-quotient rings, Abh. Math. Sem. Univ. Hamburg,37(1972)200-203.

10. B. Davvaz, V. Leoreanu, Hyperring Theory and Applacations, International Academic Press, 2007,

p. 8.

11. B. Davvaz, T. Vougiouklis , n-ary hypergroups, Iran.J. Sci. Technol. Trans. A. Sci. 30(A2)(2006)

165-174.

12. W. Dornte, Untersuchungen Uber einen verallgemeinerten Gruppenenbegri,Math. Z. 29(1928) 1-

19.

13. V. Leoreanu, Canonical n-ary hypergroups, Ital. J. Pure Appl. Math. 24(2008).

14. V. Leoreanu-Fotea, B. Davvaz, n-hypergroups and binary relations, European J. Combin. 29(2008)

1027-1218.

15. V. Leoreanu-Fotea, B. Davvaz, Roughness in n-ary hypergroups, Inform. Sci. 178(2008) 4114-

4124.

16. F. Marty, Sur une generalization de group in: 8iem congres des Mathematiciens Scandinaves,

Stockholm. 1934, pp. 45-49.

17. S. Mirvakili, B. Davvaz, Constructions of (m; n)-hyperrings. MATEMAT. 67,1(2015)1-16.

18. S. Mirvakili, B. Davvaz, Relations on Krasner (m; n)-hyperrings. European J. Combin. 31(2010)

790-802.

19. H. Shojaei, R. Ameri, Some Results On Categories of Krasner Hypermodules, J. Fundam. Appl.

Sci. 2016, 8(3S), 2298-2306.

20. T. Vougiouklis, Hyperstructure and their Representations, Hardonic, Press, Inc, 1994.

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1Department of Mathematics and Computer Sciences, University of Udine, Udine, Italia


Hyperstructures and some of the most recent applications

Piergiulio Corsini1

After a brief history of Hypergroups, since the beginning around the 40s till

today, one gives an excursus of the most recent applications of this topic to Fuzzy Sets and

Chinese groups as HX-hypergroups.

Keywords: Hyperstructures


References:

1. P.Corsini, Prolegomena of Hypergroup Theory, Aviani Editore (1993), pp. 216.

2. P.Corsini, Algebra per Ingegneria, Aviani Editore, (1991)

3. P.Corsini, Introducere in Theoria Hipergrupurilor, Translated by V.Leoreanu, Editura

Universitatii “Al. I. Cuza” Iasi, (1998) pp.158

4. P.Corsini and V. Leoreanu, Applications of Hyperstructure Theory, Advances in

Mathematics, vol. 5, Kluwer Academic Publishers, (2003).

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1Department of Mathematics, Shahid Bahonar University, Kerman, Iran ,

2Department of Mathematics, Graduate University of Advanced Technology, Mahan-Kerman, Iran,

E-mail(s): [email protected]

Weak Closure operations on ideals of a BCK-algebra

Hashem Bordbar1 and Mohammad Mehdi Zahedi

2

Weak closure operation, which is more general form than closure operation,on ideals

of BCK-algebras is introduced, and related properties are investigated. Regarding weak

closure operation, finite type, (strong) quasi-primeness, tender and naive are considered.

Using a weak closure operation “cl” and an ideal A of a lower BCK-semilattice X with the

greatest element 1, a new ideal K of X containing the ideal Acl of X is established. Using this

ideal K, a new function

clt : I(X) → I(X); A →K

is given, and related properties are considered. We show that if “cl” is a tender (resp., naive)

weak closure operation on I(X), then so are “clt” and “clf ”.

Keywords: closure operation, (finite type, tender, naive) weak closure operation,

zeromeet element, meet ideal.

2010 AMS Classification: 06F35, 03G25.

References:

[1] H. Bordbar and M. M. Zahedi, Semi-prime closure operations on BCK-

algebra, Commun. Korean Math. Soc. 30 (2015), no. 5, 385–402.

[2] H. Bordbar and M. M. Zahedi, A finite type of closure operations on

BCK-algebra, Appl. Math. Inf. Sci. Lett. 4 (2016), no. 2, 1–9.

[3] Y. Huang, BCI-algebra, Science Press, Beijing 2006.

[4] J. Meng and Y. B. Jun, BCK-algebras, Kyung Moon Sa Co. 1994.

9



1Department of Mathematics, Al al-Bayt University, Al Mafraq, Jordan 2Department of Mathematics, Al al-Bayt University, Al Mafraq, Jordan

E-mails:[email protected], [email protected]

EXTENSION OF POLYGROUPS BY POLYGROUPS VIA FACTOR POLYGROUPS

Lumnije Shehu1, Hani Khashan

2

The idea of constructing extensions of polygroups via factor polygroups comes from

an extension that De Salvo introduced in [13] which is called ( , )H G -hypergroups.

Basically, given a hypergroup ( , )H and mutually disjoint sets { }i i GA where G is a given

group such that 0A H . Set ii G

K A

and define a hyper operation on K as follows: For

all ,x y H , x y x y . For all ix A and jy A such that i jA A H H , kx y A

where i j k . This extension of G by H represents a hypergroup. The wreath product

[ ]H G introduced in [2] can be obtained by De Salvo’s construction when H and G are

polygroups, 0A H and { }iA i for 0i . In our construction, we consider two polygroups

H and L . We restrict the cardinalities of sets iA , 0i to be equal to the cardinality of

some factor polygroup /H I and the cardinality of 0A equals to that of H . The hyper

operation on ii L

K A

is based on the hyper operations on the factor polygroup /H I and the

polygroup L . In principle, the element zero of L is enlarged by the polygroup H and the

rest of the elements of L are enlarged by isomorphic copies of the factor polygroup /H I .

This construction yields a polygroup in the case when the subpolygroup I is normal.

However, the kernel of a strong homomorphism is not necessarily normal, [10]. Therefore, by

weakening the condition of normality, we obtain the utmost possible extensions. Indeed, we

define and study regularly normal subpolygroups. After introducing the isomorphism

theorems subject to these subpolygroups, we are able to present our new extension via factor

polygroups.

Keywords: hypergroups, polygroups, polygroups extensions, regularly normal

subpolygroups.


References:

[1] M. Alp, B. Davvaz, Crossed polymodules and fundamental relations, U.P.B. Sci. Bull.,

Series A, 77(2): 129-140, 2015.

10



1Department of Mathematics, Al al-Bayt University, Al Mafraq, Jordan 2Department of Mathematics, Al al-Bayt University, Al Mafraq, Jordan


[2] S. D. Comer, Extension of polygroups by polygroups and their representations using

colour schemes, Lecture notes in Math., 1004: 91-103, 1982.

[3] S. D. Comer, A remark on chromatic polygroups, Congr. Numer., 38: 85-95, 1983.

[4] S. D. Comer, Constructions of color schemes, Acta Univ. Carolin. Math. Phys., 24: 39-48,

1983.

[5] S. D. Comer, Some problems on hypergroups, Fourth Int. Con. on AHA, 67-74, 1990.

[6] S. D. Comer, Combinatorial aspects of relations, Algebra Universalis., 18: 77-94, 1984.

[7] P. Corsini, Prolegomena of Hypergroup Theory, Second edition, Aviani Editore, 1993.

[8] P. Corsini, V. Loreanu, Application of Hyperstructure Theory, Kluwer: Academic

Publishers, 2003. [9] B. Davvaz, On polygroups and permutation polygroups, Math.

Balkanica (N.S.), 14: 41-58, 2000. [10] B. Davvaz, Isomorphism theorems of polygroups,

Bull. Malays. Math. Sci. Soc., 33(2): 385-392, 2010.

[11] B. Davvaz, Polygroup Theory And Related Systems, World Scientific Publishing Co.,

2013.

[12] B. Davvaz, V. Leoreanu-Fotea, Hyperring theory and applications, International

Academic Press, Palm Harbor, Fla, USA, 2007.

[13] M. De Salvo, Gli (H,G)-ipergruppi, Riv. Mat. Univ. Parma, 10: 207-216, 1984.

[14] M. De Salvo, G. Lo Faro, On the n*-complete hypergroups, Discrete Mathematics,

208/209: 177-188, 1999. 177-188.

[15] M. Dresher and O. Ore, Theory of Multigroups, Amer. J. Math., 60: 705-733, 1938.

[16] J. Jantosciak, Homomorphisms, equivalences and reductions in hypergroups, Riv. Mat.

Pura Appl., 9: 23-47, 1991.

[17] C. G. Massouros, Some properties of certain subhypergroups, Ratio Mathematica, 25:

67-76, 2013.

[18] M. Tallini, Hypergroups and geometric spaces, Ratio Mathematica, 22: 69-84, 2012.

[19] T. Vougiouklis, Hv-groups defined on the same set, Discrete Math., 155: 259-265, 1996.

11

13th AlgebraicHyperstructuresandits Applications (AHA2017)


1Mohammad Ali Dehghani Department of Mathematics, Yazd University, Yazd, Iran 2Bijan Davvaz Department of Mathematics, Yazd University, Yazd,Iran 3Murat Alp Department of Mathematics, Nigde Ömer Halisdemir University, Nigde, Turkey

E-mail(s): [email protected], [email protected], [email protected]

On crossed polysquares and fundamental relations

Mohammad Ali Dehghani1, Bijan Davvaz

2, Murat Alp

3

In this paper, we introduce the notion of crossed polysquare of polygroups and we give some

of its properties. Our results extend the classical results of crossed squares to crossed

polysquares. One of the main tools in the study to polygroups is the fundamental relations.

These relations connevt polygroups to groups, and on the other hand, introduce new important

classes. So, we consider a crossed polysquare and by using the concept of fundamental

relation, we obtain a crossed square.

Keywords: Crossed module, crossed square, polygroup, fundamental relation.

2010 AMS Classification: 13D99, 20N20, 18D35

References:

1- M. Alp, Actor of crossed modules of algebroids, Proc. 16th Int. Conf. Jangjeon Math.

soc.,16(2005) 6-15.

2- M. Alp, Pullback crossed modules of algebroids, Iranian J. Sci. Tech.,Transaction A,

32(A3) (2008) 145-181.

3- M.~Alp, Pullbacks of profinite crossde modules and cat 1-profinite groups, Algebras

Groups Geom, 25 (2) (2008) 215-221.

4- M. Alp and B. Davvaz, On Crossed Polymodules and Fundamental Relations, U.P.B. Sci.,

Bull., Series A, 77 (2) (2015) 129-140.

5- M. Alp and Ö. Gürmen, Pushouts of profinite crossed modules and cat 1-profinite groups,

Turkish Journal of Mathematics, 27 (2003) 539-548.

6- Z. Arvasi , Crossed squares and 2-crossed modules of commutative algebras, Theory and

Applications of Categories, 3 (7) (1997) 160-181.

7- Z. Arvasi and T. porter, Freeness conditions for 2-crossed modules of commutative

algebras, Applied Categorical Structures, (to appear).

8- Z. Arvasi and E. Ulualan, On algebraic models for homotopy 3-types, Journal of

Homotopy and Related Structures, 1 (1) (2006) 1-27.

9- Z. Arvasi and E. Ulualan, 3-Types of simplicial groups and braided regular crossed

modules, Homotopy and Applications, 9 (1) (2007) 139-161.

10- H. J. Baues, Combinatorial Homotopy and 4-Dimensional Compexes, Walter de Gruyter,

Berlin, De Gruyter expositions in Mathematics, (1991).

11- R. Brown , Computing Homotopy types using crossed N-cubes of groups, Adams

Memorial Symposium on Algebraic Topology, 1 (1992) 187-210.

12- R. Brown and N. D. Gilbert, Algebraic models of 3-types and automorphism structures

12








for crossed modules, Proc. London Math. Soc. 59 (3) (1989) 51-73.

13- R. Brown and J. L. Loday, Van Kampen theorems for diagrams of spaces, Topology, 26

(1987) 311-334.

14- R. Brown and G. H. Mosa, Double categories, R-categories and crossed modules, U. C.

N. W maths preprint 88 (11). (1988) 1-18.

15- P. Carrasco , A. M. Cegarra and A. R. Garzòn, The classifying space of categorical

crossed module, Mathematische Nachrichaten, 283 (4). (2010) 544-567.

16- S. D. Comer, Polygroups derived from cogroups, J. Algebra, 89 (1984) 397-405.

17- D. Conduchè, Modules crois\ès gènèralisès de longueur 2, J. Pure Applied Algebra 34

(1984) 155-178.

18- D. Conduchè, Simplicial crossed modules and mapping cones, Georgian Math. J. 10 (4)

(2003) 623-636.

19- P. Corsini, Prolegomena of hypergroup theory, Second edition, Ariain editor(1993).

20- B.Davvaz, A survey on polygroups and their properties,Proceedings of the International

Confrence on Algebra, (2010) 148-156, Sci. Publ., Hackensack, NJ, (2012).

21- B.Davvaz, Applications of the $r^*$-relationPolygroup theory and related systems, World

Sci. Publ., (2013).

22- B. Davvaz, On Polygroups and Permutation Polygroups, Math., Balkanica (N. S.),14 (1-2)

(2000) 41-58.

23- B. Davvaz, Isomorphism theorems of polygroups, Bulletin of the Malaysian Mathematical

Sciences Society (2), 33 (3) (2010) 385-392.

24- B. Davvaz, Polygroup theory and related systems, World Sci. Publ., 2013.

25- D.Freni, A note on the core of a hypergroup and the transitive closure 𝛽∗ of 𝛽, Riv. Math.

Pura Appl., 8 (1991) 153-156.

26- D. Guin-Walery and J. L. Loday, Obstructionà 1'excision en k-thèories algèbrique, In

Friedlander, E. M., Stein, M. R. (eds.)Evanston conf. On Algebraic k-Theory (1980). (Lect.

Notes Math. 854) Springer, Berlin, Heidelberg, New York (1981) 179-216.

27- F. J. Korkes and J. Porter, Profinite crossed modules,U. C. N. W pure mathematics

preprint 86 (11). (1986).

28- M. Koskas, Groupoids, demi-groups et hypergroups, J. Math. Pures Appl., 49 (1970)

155-192.

29- V. Leoreanu-Fotea, The heart of some important classes of hypergroups, Pure Math.

Appl., 9 (1998)351-360.

30- J. L. Loday, Spaces with finitely many non-trivial homotopy groups, J. Appl.~ Algebra

24 (1982) 179-202.

31- K. Norrie, Actions and automorphisms of crossed modules, Bull. Soc. Math. France,

(1990) 118.,129-146.

32- J. Porter, N-Types of Simplicial Groups And Crossed N-Cubes, Topology,~32 (1).(1993)

5-24.

13








33- T. Vougiouklis, Hyperstructures and their representations, Hadronic Press, Inc, 115, Palm

Harber, USA (1994).

34- J. H. C.Whitehead, Combinatorial homotopy II, Bull. Amer. Math. Soc. 55 (1949) 453-

496.

14






1Brno University of Technology, Faculty of Electrical Engineering and Communication, Brno, Czech Republic 2Masaryk University, Faculty of Economics and Administration, Brno, Czech Republic

E-mail(s): [email protected], [email protected],

Recent Advances in EL-hyperstructures

Michal Novák1, Štěpán Křehlík

2

EL-hyperstructures are semihypergroups or ring-like hyperstructures S constructed

from partially (or, in many cases, quasi-) ordered semigroups, where the hyperoperation on S

is defined by 𝑎 ∗ 𝑏 = [𝑎 . 𝑏)≤ = {𝑥 ∈ 𝑆 |𝑎 . 𝑏 ≤ 𝑥} for all 𝑎, 𝑏 ∈ 𝑆 . When looking for

examples, one can construct numerous EL-semihypergroups, hypergroups, join spaces,

lattice-like or ring-like hyperstructures in a number of natural contexts as the set S can be a

number domain, set of words of a given alphabet, set of objects, properties of which can be

described by numbers, sets of vectors or matrices, etc. The relations can be numerous as well:

ordering numbers (or numerically described properties) by size, divisibility relation, or

relations motivated by some special contexts. EL-hyperstructures were introduced by

Chvalina in [1] and named so and studied by Novák in e.g. [2,3].

In our paper we focus on some recent advances in the area of EL-hyperstructures. We

clarify the issue of antisymmetry of the relation ``≤" and include examples when it is a quasi-

ordering which is moreover symmetric, i.e. an equivalence. We show the use of the

construction in the area of lattice-like hyperstructures, i.e. for 𝐻𝑣 -semilattices,

hypersemilattices or hyperlattices (which were studied in [4]). We discuss implications of

extensivity of the hyperoperation, i.e. contexts when {𝑎, 𝑏} is included in 𝑎 ∗ 𝑏 for all 𝑎, 𝑏 ∈

𝑆 . We also mention the way EL-hyperstructures can be used to construct Cartesian

composition of multiautomata. Finally, we briefly mention the relation of EL-hyperstructures

to some other concepts of hyperstructure theory, where the idea of ordering is used, such as

ordered hyperstructures, quasi-order hypergroups or some special cases of BCI-algebras.

Keyword(s): EL-hyperstructures, hyperstructure theory, ordered semigroups, quasi-ordered

semigroup

2010 AMS Classification: 20N20, 06F05

Reference(s):

1. Chvalina J., Functional Graphs, Quasi-ordered Sets and Commutative Hypergroups,

Masaryk University, Brno, 1995. (in Czech)

2. Novák M., Some basic properties of EL-hyperstructures, European J.~Combin., 34, 446-

459, 2013.

3. Novák M., On EL-semihypergroups, European J. Combin., 44(Part B), 274-286, 2015.

4. Křehlík Š., Novák M., From lattices to 𝐻𝑣-matrices, An. St. Univ. Ovidius Constanta,

24(3), 209-222, 2016.

15



1Mersin University, Department of Mathematics, Mersin, Turkey


HH*-Intuitionistic Heyting Valued -Algebra

Sinem Tarsuslu(Yılmaz)1, Gökhan Çuvalcıoğlu

1

Intuitionistic Logic was introduced by L. E. J. Brouwer and Heyting algebra was

defined by A. Heyting in 1930, to formalize the Brouwer’s intuitionistic logic. The concept of

Heyting algebra has been accepted as the basis for intuitionistic propositional logic. Heyting

algebras have had applications in different areas. The co-Heyting algebra is the same lattice

with dual operation of Heyting algebra. Also, co-Heyting algebras have several applications

in different areas.

In this paper, we introduced the new concept HH*- Intuitionistic Heyting Valued -

Algebra. The purpose of introducing this new concept is to expand the field of researchers’

area using both membership degree and non-membership degree. This allows us to get more

sensitive results.The concepts of HH*- Intuitionistic Heyting valued set, HH*- Intuitionistic

Heyting valued relation, HH*- Intuitionistic Heyting valued -algebra and the

homomorphism over HH*- Intuitionistic Heyting valued -algebra were defined.

Keyword(s): Heyting Valued Algebra, co-Heyting Valued Algebra, Omega Algebra,

Intuitionistic Logic.

2010 AMS Classification: 03C05

Reference(s):

1. Brouwer, L. E. J., Intuitionism and Formalism, English translation by A. Dresden, Bulletin

of the American Mathematical Society, 20 (1913): 81--96, reprinted in Benacerraf and

Putnam (eds.) 1983: 77--89; also reprinted in Heyting (ed.) 1975: 123--138.

2. Çuvalcığlu G., Heyting Valued Omega Free Algebra, Çukurova University Institute of

Science and Technology, PhD Thesis, Adana, 2002, 68 p.

3. Faith C., Algebra: Rings, Modules and Categories I, Springer-Verlag, Berlin

4. Heyting, A. "Die formalen Regeln der intuitionistischen Logik," in three parts,

Sitzungsberichte der preussischen Akademie der Wissenschaften: 42--71, 158--169, 1930.

English translation of Part I in Mancosu 1998: 311--327.

5. Lawvere F.V. , Intrinsic co-Heyting boundaries and the Leibniz rule in certain toposes, in

A.Carboni, et.al., Category theory, Proccedings, Como. 1990.

This study was supported by the Research Fund of Mersin University in Turkey with Project Number:

2015-TP3-1249.

16




E-mail(s):[email protected], [email protected]

Algebraic Approach to Multiplicative Set

Sinem Tarsuslu(Yılmaz)1, Gökhan Çuvalcıoğlu

1

Rough set theory was introduced by Pawlak in 1982. The theory of rough sets is an

extension of set theory as a subset of a universe is defined by a pair of ordinary sets called the

lower and upper approximations.The algebraic structures of rough sets were studied by

several authors.Davvaz introduced the notion of rough subring in 2004. Some properties of

the lower and the upper approximations in a ring were examined by Davvaz.

In this study, we examined some relations between rough sets and multiplicative

subsets of a commutative ring R. The lower and upper approximations of the set X with

respect to " " were defined and algebraic properties were examined.

Keyword(s): Rough set,Rough subring, Multiplicative set, Fuzzy ideal.

2010 AMS Classification: Primary 05C38, 15A15; Secondary 05A15, 15A18

References:

1. Bonikowaski Z., Algebraic structures of rough sets, in: W.P.Ziarko (Ed.), Rough Sets

Fuzzy Sets, and Knowledge Discovery, springer-Verlag, Berlin, (1995), pp.242-247.

2. Biswas R., Nanda S., Rough groups and rough subgroups, Bull. Polish Acad. Sci. Math. 42

(1994), 251-254.

3. Corsini P., Rough sets, fuzzy sets and join spaces, Honorary volume dedicated to Prof.

Emeritus J.Mittas, Aristotle Univ. of Thessaloniki, (1999-2000).

4. Davvaz B., Roughness based on fuzzy ideals, Inform. Sci.176 (2006), 2417-2437.

5. Davvaz B., Rough sets in a fundamental ring, Bull. Iranian Math. Soc. 24 (1998), 49-61.

6. Dubois D., Prade H., Rough fuzzy sets and fuzzy rough sets, Int. J.General Syst. 17 (2-3)

(1990), 191-209.

7. Iwinski T., Algebraic approach to rough sets, Bull. Polish Acad. Sci. Math. 35 (1987), 673-

683.

8. Kuroki N., Rough ideals in semigroups, Inform. Sci. 100 (1997), 139-163.

9. Kuroki N., Mordeson J.N., Structure of rough sets and rough groups, J.Fuzzy Math. 5 (1)

(1997), 183-191.

10. Kuroki N., Wang P.P., The lower and upper approximations in a fuzzy group, Inform. Sci.

90 (1996) 203-220.

11. Mordeson J.N., Rough set theory applied to (fuzzy) ideal theory, Fuzzy Sets and Systems

121 (2001), 315-324.

12. Pawlak Z., Rough sets, Int. J.Comput. Inform. Sci. 11 (1982), 341-356.

17





13. Pawlak Z., Rough Sets-Theoretical Aspects of Reasoning about Data, Kluwer Academic

Publishers, Dordrecht, (1991).

14. Pawlak Z., Skowron A., Rough sets: some extensions, Inform. Sci. 177 (1) (2007), 28-40.

15. Pomykala J., Pomykala J.A., The stone algebra of rough sets, Bull. Polish Acad. Sci.

Math. 36 (1988) 495-508.

16. Sarkar M., Rough-fuzzy functions in classification, Fuzzy Sets Syst. 132 (2002) 353--369.

17. W. Liu, Operations on fuzzy ideals, Fuzzy Sets and Systems 8(1983), 31-41.

18. Zadeh L.A., Fuzzy Sets, Information and Control, 8, (1965), p. 338-353.

19. Zadeh L.A., The concept of linguistic variable and its applications to approximate

reasoning,

Part I, Inform. Sci. 8 (1975) 199-249;

Part II, Inform. Sci. 8 (1975) 301-357;

Part II, Inform. Sci. 9 (1976) 43-80;

This study was supported by the Research Fund of Mersin University in Turkey with Project Number:

2015-TP3-1249.

18



(1) Department of Mathematics, Payamnour University. P.O. Box 19395-3697, Tehran, Iran.

(2) School of Mathematics, Statistics and Computer Science, Collage of Sciences, University of Tehran,

P.O. Box 14155-6455, Tehran, Iran.

[email protected], [email protected]

On the relation between categories of ),( nm -ary hypermodules and ),( nm -ary modules

N .Jafarzadeh(1)

,R. Ameri(2)

‎ We introduce the category of ),( nmR -hypermodules over a Krasner ),( nm -hyperring R and

obtain some categorical objects in this category such as product and coproduct. We apply the

fundamental relations * and

* on, M and, R respectively to construct fundamental functor from

the category of ),( nmR -hypermodules into category of */R -modules. In particular we consider the

fundamental relation on ),( nm -hypermodules, and construct functor from the category of ),( nm -

hypermodules to the category of ),( nm -modules. Then, we find the relations between hom, product,

coproduct and fundamental functor.

‎In this section we give some definitions and results of n-ary hyperstructures which we need in

what follows.

‎A mapping 𝑓:𝐻 × 𝐻 × …× 𝐻⏟ 𝑛

→ 𝑃∗(𝐻) is called an n-ary hyperoperation, where 𝑃∗(𝐻)is the

set of all non-empty subsets of H‎.

‎ Definition 2.3 [11] A Krasner ),( nm -hyperring is algebraic hyperstructure ),,( khR

which satisfies the following axioms:

• ),( hR is a canonical m -ary hypergroup;

• ),( kR is an n -ary semigroup;

• the n -ary operation k is distributive to the m -ary hyperoperation ,h i.e, for all

,,, 11

1

1 Rxaa mn

i

i

and ,1 ni

));,,(,),,,((=)),(,( 1

1

111

1

111

1

1

n

im

in

i

in

i

mi axakaxakhaxhak

• 0 is a zero element (absorbing element), of the n -ary operation ,., eik for Rxn 2 we

hav:e

,0).(==),0,(=)(0, 2322

nnn xkxxkxk

‎ Definition 2.5 [3] A Krasner ),( nm -hypermodule ),,( gfM is an ),( nm -hypermodule

with a canonical m -ary hypergroup ),( fM over a Krasner ),( nm -hyperring ).,,( khR

‎ Definition 2.8 [11] Let ),,( khR be ),( nm -hyperring.The relation * is the smallest

equivalence relation such that the quotient )/,/,/( *** khR be ),( nm -ring. where */R is the set

of equivalence classes. The * is called fundamental equivalence relation.

19







Various categories of (m‎, ‎n) -ary hypermodules‎

‎‎ Definition 3.2 The category HmodR nm ),( of ),( nm -ary hypermodules defined as follows:

• the objects of HmodR nm ),( are ),( nm - hypermodules,

• for the objects M and ,K the set of all morphisms from M to K is defined as follows:

},smhomomorphiman is )(:|{=),( * KPMffKMHomR

• the composition gf of morphisms )(: * KPMf and )(: * LPKg defined as

follows:

),(=)( ),(:)(

* tgxgfKPHgfxft

• for any object ,H the morphism ),(:1 * HPHH defined by },{=)(1 xxH is the

identity morphism.

Theorem 3.5 modRhmodR nmnms *

),(),( /:F defined by */=)( MMF and

,=)( *F is a functor ,//: and : *

2

*

1

*

21 MMMM where modR nm *

),( / is the

category of all ),( nm -modules over ./ *R

Remark 3.9 In the following of this paper we consider the category of all ),( nm -

hypermodules over a ),( nm -hyperring ,R in the sense of Krasner ),( nm -hypermodules over

commutative Krasner ),( nm -hyperring R with identity. We denote this category by

.),( KHmodR nm Hence, the objects of KHmodR nm ),( are Krasner ),( nm -hypermodules over

commutative Krasner ),( nm -hyperring with identity and all morphisms are multivalued

homomorphisms.

In this section, concepts of direct hyper product and direct hyper coproduct of a Krasner

),( nm -hypermodule are defined. Also we give some properties of the category .),( KHmodR nm

Definition 4.3 Let }|{ IiM i be a family of ),( nm -hypermodules. we define a

hyperoperation on i

Ii

M

as follows:

.}{ )(|}{=}{ 111

i

Ii

im

i

im

iiii

im

i MaafttaF

20







For Rr and ,i

Ii

i Ma

define

.),(=)}{()(

1)(

1)(

1)(

1 Iii

n

iIii

n argarG

then ,i

Ii

M

together with m -ary hyperoperation F and n -ary operation G is called direct hyper

product }.|{ IiM i

Definition 4.6 The direct hyper sum of the family }|{ IiM i of ),( nm -hypermodules,

denoted by i

Ii

M

is the set of all ,}{ Iiia where ia can be non-zero only for a finite number of

indices.

Theorem 4.11 Let }|{ IiM i be a family of ),( nm -hypermodules over an ),( nm -ary

hyperring R and let Iii

M ,* and )( **

iM

Ii

iM

Ii

be fundamental equivalence relation on iM and

)( i

Ii

i

Ii

MM

respectively. then

• ,/)/(: **

1i

Mi

IiiM

Ii

i

Ii

MM

• ./)/(: **

2i

Mi

IiiM

Ii

i

Ii

MM

Theorem 4.12 Fundamental functor F preserves zero object, product and coproduct.

Proposition 5.3 Let }|{ IiM i be a family of ),( nm -hypermodules over an ),( nm -ary

hyperring R and N also is an ),( nm -hypermodule and F be fundamental functor . Then

))).,((()),(( NMhomNMhom iR

Ii

i

Ii

R FF

Corollary 5.5 Let CBA ,, be ),( nm -hypermodules over an ),( nm -ary hyperring R and

and F be fundamental functor . Then isomorphism

)),(()),(()),(( CBhomCAhomCBAhom RRR FFF

is natural.

21







Keywords: category, ),( nm -hypermodules, product, coproduct, additive category.

References

[1] R. Ameri, On the categories of hypergroups and hypermodules, J. Discrete Math. Sci.

cryptogr. 6 (2003) 121-132.

[2] R. Ameri, M.Norouzi, Prime and primary hyperideales in Krasner ),( nm -hyperring,

European J. Combin. 34(2013)379-390.

[3] SM. Anvariyeh, S. Mirvakili, B. Davvaz, Fundamental relation on ),( nm -hypermodules

over ),( nm -hyperrings. Ars combin. 94(2010)273-288.

[4] P. Corsini, Prolegemena of Hypergroup Theory, second ed. Aviani, Editor, 1993.

[5] B. Davvaz, V. Leoreanu, Hyperring Theory and Applacations, International Academic

Press, 2007, p. 8.

[6] B. Davvaz, T. Vougiouklis , n -ary hypergroups, Iran.J. Sci. Technol. Trans. A. Sci.

30(A2)(2006) [7] W. Dörnte, Untersuchungen Über einen verallgemeinerten Gruppenenbegriff, Math.

Z. 29(1928) 1-19.

[8] V. Leoreanu, Canonical n -ary hypergroups, Ital.J. Pure Appl. Math. 24(2008).

[9] F. Marty, Sur une generalization de group in: iem8 congres des Mathematiciens

Scandinaves, Stockholm. 1934, pp. 45-49.

[10] S. Mirvakili, B. Davvaz, Constructions of ),( nm -hyperrings. MATEMAT.

67,1(2015)1-16.

[11] S. Mirvakili, B. Davvaz, Relations on Krasner ),( nm -hyperrings. European J. Combin.

31(2010) 790-802.

[12] H. Shojaei, R. Ameri, Some Results On Categories of Krasner Hypermodules. J

FundamAppl Sci. 2016, 8(3S), 2298-2306.

[13] T. Vougiouklis, Hyperstructure and their Representations, Hardonic, Press, Inc, 1994.

22



1Ordu University, Department of Mathematics, Ordu, Turkey

E-mails:[email protected]

Soft Bi-Ideals of Soft LA-Semigroups

Yıldıray Çelik1

In this paper, we present notion of soft bi-ideal of a soft ring and give some results on

it. Also, we introduce concept of soft bi-ideal of a soft LA-semigroup and investigate some

properties of it.

Keywords: soft ring, soft bi-ideal, soft LA-semigroup

2010 AMS Classification: 06F05, 16D25

Reference(s):

1. Acar U., Koyuncu F., Tanay B., Soft sets and soft rings, Comput. Math. Appl., 59, 3458-

3463, 2010.

2. Aktaş H., Çağman N., Soft sets and soft groups, Inform. Sci., 177, 2726-2735, 2007.

3. Ali M.I., Shabir M., Naz M., Algebraic structures of soft sets associated with new

operations, Comput. Math. Appl., 61(9), 2647-2654, 2011.

4. Aslam M., Shabir M., Mehmood A., Some studies in soft LA-semigroups, J. Adv. Res.

Pure Math., 3(4), 128-150, 2011.

5. Çelik Y., Ekiz C., Yamak S., A new view on soft rings, Hacet. J. Math. Stat., 40(2), 273-

286, 2011.

6. Feng F., Jun Y.B., Zhao X., Soft semirings, Comput. Math. Appl., 56, 2621-2628, 2008.

7. Maji P.K., Biswas R., Roy A.R., Soft set theory, Comput. Math. Appl., 45, 555-562, 2003.

8. Molodtsov D., Soft set theory-first results, Comput. Math. Appl., 37, 19-31, 1999.

23



1Brno University of Technology, Faculty of Electrical Engineering and Communication, Brno, Czech 2 Brno University of Technology, Faculty of Electrical Engineering and Communication, Brno, Czech 3 Brno University of Technology, Faculty of Electrical Engineering and Communication, Brno, Czech

E-mail(s):[email protected], [email protected], [email protected]

Sequences of Groups and Hypergroups of Linear Ordinary Differential Operators

Jan Chvalina1, Michal Novák

2, David Staněk

3

Linear ODE's are a classical tool for constructing many useful models for description

of numerous processes. Given their standard forms, left-hand sides of such equations (both of

homoegeneous and non-homogeneous) are called linear differential operators. Groups of such

operators of different orders can be constructed and some of their properties studied. This

includes solvability, relation to quasi-automata or actions on specific spaces or structures.

Using a suitable ordering or quasi-ordering of groups of linear differential operators

we construct hyperstructures of linear differential operators. Then, with the help of these

hyperstructures, we construct multiautomata which are cardinal sums of perfect semisimple

submultiautomata.

The main objective of our paper is to focus on the study of sequences (finite or

countable) of groups and hypergroups of linear differential operators of decreasing orders. For

this we use inclusion embeddings and group homomorphisms. In particular, we obtain and

study what we call ``coupled sequences". We also include a construction of sequences of

second-order linear differential operators in the Jacobi form, i.e. such operators that the

coefficient at the first-order derivative is zero. Our results can be generalized to operators of

an arbitrary order.

We also apply our considerations to the theory of (multi-)automata. In particular, we

obtain actions of abelian groups and hypergroups constructed from linear spaces of

polynomials of various dimensions over additive abelian groups of differential operators of

the corresponding order with constant coefficients at the highest-order derivatives.

Keyword(s): hyperstructure theory, linear differential operators, ODE, theory of automata

2010 AMS Classification: 20N20, 68Q70, 47D03

Reference(s):

1. Bavel Z., The source as a tool in automata, Information and Control, 18, 140-155, 1971.

2. Jan J., Digital Signal Filtering, Analysis and Restoration, IEEE Publ. London, 2000.

3. Gécseg F., Péak I., Algebraic Theory of Automata, Budapest, Akadémia Kiadó, 1972.

4. Chvalina J., Novák M. and Křehlík Š., Cartesian composition and the problem of

generalising the MAC condition to quasi-multiautomata. An. St. Univ. Ovidius Constanta,

24(3), 79-100, 2016

5. Chvalina J., Staněk, D. Sequences of automata formed by groups of polynomials and by

semigroups of linear differential operators, in: 16th Conference on Applied Mathematics

APLIMAT 2017. Proceedings. SUT in Bratislava, 2017.

24

On Different Approach of Fuzzy Ring Homomorphims (AHA2017)


1Recep Tayyip Erdogan University, Department of Mathematics, Rize, Turkey

E-mail(s):[email protected]

On Different Approach of Fuzzy Ring Homomorphims

Umit DENİZ1

In this study we approach the definition of TLring homomorphism. In literature the

definition of fuzzy ring homomorphism is given by using the classic functions. In this study

we give the definition of fuzzy ring homomorphism by using the definition of Mustafa

Demirci’s fuzzy function. Some definition and theorems of ring homomorphism in classic

algebra is adapted to fuzzy algebra and proved.

Keywords: Fuzzy Functions, Fuzzy Equivalence Relations, Triangular Norms, Fuzzy

Subrings, Fuzzy Ideals

2010 AMS Classification: Mathematics

References:

1. Zadeh L.A., Fuzzy sets, Inform. and Control, 8, 338-353, 1965.

2. Rosenfeld A., Fuzzy groups, J.Math Anal Appl., 35 , 512-517, 1971.

25



1 University of Bojnord, Iran E-mail: [email protected]

𝓛-hyperstructures

Mahmood Bakhshi1

In this paper, as a generalization of familiar classical ordered algebraic structures such

as ordered semigroups and ordered groups the notion of 𝓛 -hyperstructure is introduced.

Giving some examples it is shown that the familiar ordered hyperstructures and also those

hyper algebraic structures arose from logic can be viewed as a special types of 𝓛 -

hyperstructres. After that investigating basic properties, some types of hyperideals are

introduced, thier properties are investigated and some characterizations and the connections

among them are obtained.

Keyword(s): hyperstructure, ordered sets, algebras of logics

2010 AMS Classification: 20N20, 06F15, 06F35, 06D35

1. Main results

Definition 2.1. By a language of hyperstructures we mean a set 𝓛 consists of a set ℛ of

relation symbols and a set ℱ of set-valued function symbols such that to each member of ℛ (

of ℱ) is associated a natural number (a non-negative integer) called the arity of the symbol.

ℱ𝑛 denotes the set of set-valued function symbols in ℱ of arity 𝑛, and ℛ𝑛 denotes the set of

relation symbols in ℛ of arity 𝑛.

Definition 2.2. Let 𝓛 be a language. An ordered pair 𝑨 =< 𝐴; 𝓛 > in which 𝐴 is a non-empty

set and 𝓛 consists of a family 𝑅 of fundamental relations 𝑟𝑨 on 𝐴 indexed by ℛ and a family

ℱ of fundamental hyper operations 𝑓𝑨 on 𝐴 indexed by ℱ is called a hyperstructure of type 𝓛

(or 𝓛 -hyperstructure). 𝐴 is called the universe of 𝑨. When ℛ = ∅, 𝑨 is a hyper algebra and if

ℱ = ∅, 𝑨 is a relational structure. If 𝓛 is finite, say ℱ = {𝑓1, … , 𝑓𝑚} and ℛ = {𝑟1, … , 𝑟𝑛}, we

often write < 𝐴; 𝑓1, … , 𝑓𝑚; 𝑟1, … , 𝑟𝑛 > instead of < 𝐴; 𝓛 >. If 𝑓𝑖 ∈ ℱ𝑛 is an 𝑛𝑖-ary function

symbol and 𝑟𝑗 ∈ ℛ𝑙 is an 𝑙𝑗-ary relation we write, for brevity, ℒ<𝑛1,…,𝑛𝑚;𝑙1,…,𝑙𝑘>-hyperstructure,

where 𝑛1 ≥ 𝑛2 ≥ ⋯ ≥ 𝑛𝑚 and 𝑙1 ≥ 𝑙2 ≥ ⋯ ≥ 𝑙𝑘, instead of 𝓛 -structure. İn this paper, we

focus on ℒ<2;2>-hyperstructures; for brevity, whenever it is clear from the context, we write

ℒ-hyperstructure instead of ℒ<2;2>-hyperstructure.

Definition 2.3. Let < 𝐻;∘, ≤> be an ℒ-hyperstructure. The relation ≤ can be eneralized to

nonempty subsets of 𝐻 as follows: for 𝐴, 𝐵 ∈ 𝑃∗(𝐻),

(i) 𝐴 ≤𝑤 𝐵, if there exist 𝑎 ∈ 𝐴 and 𝑏 ∈ 𝐵 such that 𝑎 ≤ 𝑏,

(ii) 𝐴 ≤𝑟𝑤 𝐵, if for each 𝑏 ∈ 𝐵 there exists 𝑎 ∈ 𝐴 such that 𝑎 ≤ 𝑏,

26




(iii) 𝐴 ≤𝑙𝑤 𝐵, if for each 𝑎 ∈ 𝐴 there exists 𝑏 ∈ 𝐵 such that 𝑎 ≤ 𝑏,

(iv) 𝐴 ≤𝑡𝑤 𝐵, if 𝐴 ≤𝑙𝑤 𝐵 and 𝐴 ≤𝑟𝑤 𝐵,

(v) 𝐴 ≤𝑠 𝐵 if for each 𝑎 ∈ 𝐴 and 𝑏 ∈ 𝐵 we have 𝑎 ≤ 𝑏.

Definition 2.4. Let < 𝐻;∘, ≤ > be an ℒ-hyperstructure. We say that ≤ is

(i) weak left (right) compatible if

𝑥 ≤ 𝑦 ⟹ 𝑎 ∘ 𝑥 ≤𝑤 𝑎 ∘ 𝑦 ( 𝑥 ∘ 𝑎 ≤𝑤 𝑦 ∘ 𝑎) ∀𝑎 ∈ 𝐻

If ≤ is weak left and weak right compatible it is said to be weak compatible.

(ii) r-left (r-right) compatible if

𝑥 ≤ 𝑦 ⟹ 𝑎 ∘ 𝑥 ≤𝑟𝑤 𝑎 ∘ 𝑦 ( 𝑥 ∘ 𝑎 ≤𝑟𝑤 𝑦 ∘ 𝑎) ∀𝑎 ∈ 𝐻

If ≤ is r-left and r-right compatible it is said to be r-compatible.

(iii) l-left (l-right) compatible if

𝑥 ≤ 𝑦 ⟹ 𝑎 ∘ 𝑥 ≤𝑙𝑤 𝑎 ∘ 𝑦 (𝑥 ∘ 𝑎 ≤𝑙𝑤 𝑦 ∘ 𝑎) ∀𝑎 ∈ 𝐻

If ≤ is l-left and l-right compatible it is said to be l-compatible.

(iv) t-left (t-right) compatible if ≤ is l-left and r-left compatible (l-right and r-right

compatible).

If ≤ is t-left and t-right compatible it is said to be t-compatible or briefly compatible.

(v) strong left (right) compatible if

𝑥 ≤ 𝑦 ⟹ 𝑎 ∘ 𝑥 ≤𝑠 𝑎 ∘ 𝑦 (𝑥 ∘ 𝑎 ≤𝑠 𝑦 ∘ 𝑎) ∀𝑎 ∈ 𝐻

If ≤ is strong left and strong right compatible it is said to be strong compatible.

Definition 2.5. Let ≤ be any types of the relations introduced in Definition 2.3. We say that ≤

is reversed left (right) compatible if

𝑥 ≤ 𝑦 ⟹ 𝑎 ∘ 𝑦 ≤ 𝑎 ∘ 𝑥 (𝑟𝑒𝑠𝑝. 𝑦 ∘ 𝑎 ≤ 𝑥 ∘ 𝑎) ∀𝑎 ∈ 𝐻

Definition 2.6. By a (weak, l-, r-, t-, strong) ℒ<2;2>-hyperstructure we mean an ℒ-

hyperstructure on which is defined a (weak, l-, r-, t-, strong) compatible binary relation.

Remark 2.7. For convenience, we drop the prefix two-sided and so a two-sided ℒ-

hyperstructure is called an ℒ-hyperstructure.

Definition 2.8. An ℒ-hyperstructure < 𝐻;∘, ≤ > in which ∘ is commutative (associative) is

said to be a commutative ℒ-hyperstructure (resp, ℒ-semihypergroup).

Definition 2.9. An element e of an ℒ-hyperstructure < 𝐻;∘, ≤ > is called an identity if

𝑥 ∈ 𝑥 ∘ 𝑒 ∩ 𝑒 ∘ 𝑥, ∀𝑥 ∈ 𝐻.

Example 2.10.

(i) Any hyper 𝐾-algebra [] and any hyper 𝑀𝑉-algebra [] is an ℒ<2;2>-hyperstructure in

which the binary relation satisfies Definition 2.3(i).

27




(ii) Any hyper residuated lattice [] is an ℒ<2;2>-hyperstructure in which the binary

relation is weak right compatible with respect to the multiplication and weak left

compatible with respect to the residuation.

(iii) Any hyper 𝐵𝐶𝐾-algebra [] is an ℒ<2;2>-hyperstructure in which the binary relation is

reversed 𝑙-left compatible.

(iv) Any ordered semihypergroup [] is an ℒ<2;2>-hyperstructure with an 𝑙-left compatible

relation.

(v) Consider ℝ1 = [1, ∞), the set of all real numbers greater than 1, as a poset with the

natural ordering, and define 𝑥 ∘ 𝑦 to be the set of all upper bounds of {𝑥, 𝑦}. Thus

< ℝ1;∘, ≤ >, is a commutative r-ℒ<2;2>-semihypergroup with 1 as the unique

identity.

(vi) Let < 𝐺; ∗, 𝑒, ≤ > be an ordered group, and let 𝑥 ∘ 𝑦 =< {𝑥, 𝑦} >, the subgroup of 𝐺

generated by {𝑥, 𝑦}. Then < 𝐺; ∘, ≤ > is a commutative ℒ<2;2>-hyperstructure.

(vii) Let < 𝐿; ∨, ∧, 0 > be a lattice with the least element 0. For 𝑎, 𝑏 ∈ 𝐿, let 𝑎 ∘ 𝑏 =

𝐹(𝑎 ∧ 𝑏), where 𝐹(𝑥) is the principal filter generated by 𝑥 ∈ 𝐿. Then, < 𝐿; ∘ > is a

commutative r- ℒ-hyperstructure.

(viii) Let 𝐻 = {𝑎, 𝑏} be a chain with 𝑎 < 𝑏. We define a hyperoperation `∘' on 𝐻 as in

Table 1. Then, < 𝐻; ∘, ≤ > is an r- ℒ -hyperstructure, whereas it is not l- ℒ -

hyperstructure because 𝑎 ≤ 𝑏 but 𝑎 ∘ 𝑎 ≰𝑙𝑤 𝑎 ∘ 𝑏. Indeed, 𝑏 ∈ 𝑎 ∘ 𝑎 but there is not

any element 𝑥 ∈ 𝑎 ∘ 𝑏 such that 𝑏 ≤ 𝑥.

Table 1: Cayley table of Example 2.10(viii)

∘ 𝒂 𝒃

𝒂 {𝑎, 𝑏} {𝑎}

𝒃 {𝑎} {𝑎, 𝑏}

Definition 2.11. By an ordered ℒ -hyperstructure we mean an ℒ -hyperstructure in which the

binary relation is a partial ordering.

Definition 2.14. Let 𝐻 be a (weak, left, right, strong) ordered ℒ-hyperstructure. A down set 𝐼

of 𝐻 is called a

(i) left hyperideal if 𝐻𝐼 ⊆ 𝐼,

(ii) right hyperideal if 𝐼𝐻 ⊆ 𝐼.

If 𝐴 is a left and a right hyperideal, it is called a hyperideal of 𝐻.

Example 2.15.

(i) Consider the ordered ℒ -hyperstructure < 𝕫,∘, ≤ > given in Example 2.10(vi). It is

not difficult to check that the only hyper ideal of 𝕫 is itself.

28




(ii) Let 𝐻 = {𝑎, 𝑏, 𝑐} be a partially ordered set, where 𝑎 < 𝑏 and define

hyperoperation `∘' on 𝐻 as shown in Table 2. Then < 𝐻,∘, ≤ > is an ordered ℒ-

semihypergroup in which 𝐼 = {𝑎, 𝑏} is a hyperideal of 𝐻.

Table 2: Cayley table of Example 2.15(ii)

∘ a b c

a {𝑎, 𝑏} {𝑎} {𝑎}

b {𝑎} {𝑏} {𝑏}

c {𝑎} {𝑏} {𝑐}

(iii) Consider the partially ordered set 𝐻 given in part (ii). We define a hyperoperation

on 𝐻 as in Table 3. Then < 𝐻,∘, ≤ > is an ordered ℒ -semihypergroup in which

𝐼 = {𝑎, 𝑏} is a left hyperideal of 𝐻 but it is not a right hyperideal because 𝑎 ∘ 𝑐 =

{𝑎, 𝑏, 𝑐} ⊈ 𝐼. Thus 𝐼𝐻 ⊈ 𝐼$.

Table 3: Cayley table of Example 2.15(iii)

∘ a b c

a {𝑎, 𝑏} {𝑎} {𝑎, 𝑏, 𝑐}

b {𝑎} {𝑏} {𝑐}

c {𝑎, 𝑏} {𝑎, 𝑏} {𝑎, 𝑏, 𝑐}

(iv) Consider the partially ordered set 𝐻 given in part (ii), again. Then, < 𝐻,∘, ≤ > is

an ordered ℒ -semihypergroup in which the hyperoperation is given as in Table 4.

It is easy to check that 𝐼 = {𝑎, 𝑏} is a right hyperideal of 𝐻 which is not a left

hyperideal becuase 𝑐 ∘ 𝑏 = {𝑐} ⊈ 𝐼. Hence, 𝐻𝐼 ⊈ 𝐼.

(v)

Table 4: Cayley table of Example 2.15(iv)

∘ a b c

a {𝑎, 𝑏} {𝑎} {𝑎, 𝑏}

b {𝑎} {𝑏} {𝑎, 𝑏}

c {𝑎, 𝑏, 𝑐} {𝑐} {𝑎, 𝑏, 𝑐}

Reference(s):

1. Ameri R., Bakhshi M., Nematollah Zadeh S. A., Borzooei R.A., Fuzzy (strong) congruence

relations on hypergroupoids and hyper BCK-algebras, Quasigroups Related Systems, 15, 11-

24, 2007.

2. Blyth T. S., Lattices and Ordered Algebraic Structures, Springer-Verlag, London, 2005.

3. Borzooei R. A., Hasankhani A., Zahedi M. M., On hyper K-algebras, Math. Japonica 52,

113-121, 2000.

4. Corsini P., Prolegomena of hypergroup theory, 2nd edition, Aviani editor, 1993.

5. Ghorbani S., Hasankhani A., Eslami E., Hyper MV-algebras, Set-Valued Mathematics and

Applications, 1, 205-222, 2008.

29




6. Heidari D., Davvaz B., On ordered hyperstructures, U.P.B. Sci. Bull. Series A, 73 (2), 85-

96, 2000.

7. Jun Y. B., Zahedi M. M., Xin X. L., Borzooei R. A., On hyper BCK-algebras, Italian

Journal of Pure and Applied Mathematics, 10, 127-136, 2000.

8. Marty F., Sur une generalization de la notion de group, 8th congress Math. Scandenaves,

Stockholm, 45-49, 1934.

9. Zahiri O., Borzooei R. A., Bakhshi M., (Quotient) hyper residuated lattices, Quasigroups

Related Systems 20, 125-138, 2012

30



1 Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran, Iran 2 Faculty of Sciences and Modern Technologies, Graduate University of Advanced Technology, Kerman, Iran

E-mail(s): [email protected], [email protected]

New connections between hyperstructures and Graph Theory

Masoumeh Golmohamadian 1 , Mohammad Mehdi Zahedi 2

In present study, we investigate the relation of dominating sets in graphs and

hyperstructures. We introduce different hyperoperations and semihypergroups, deriving from

dominating sets and minimal dominating sets of a graph and we examine their properties.

A vertex 𝑣 in a graph 𝐺 is said to dominate itself and each of its neighbors and a set 𝐷

of vertices of 𝐺 is a dominating set of 𝐺 if every vertex of 𝐺 is dominated by at least one

vertex of 𝐷 . let 𝐺 = (𝑉, 𝐸) be a graph, 𝐻𝑖 be a dominating set and 𝐻 be the set of all

dominating sets of 𝐺. Then we define 𝜃(𝐻𝑖) as the maximum number of vertices of 𝐻𝑖, that

we can omit from 𝐻𝑖 to convert it to a minimal dominating set. For every 𝐻𝑖 , 𝐻𝑗 ∈ 𝐻, we

define the commutative semihypergroup (𝐻,∗) in the following way:

𝐻𝑖 ∗ 𝐻𝑗 =

{

𝐻𝑖 , 𝑖𝑓 𝜃(𝐻𝑖) < 𝜃(𝐻𝑗)

𝐻𝑗 , 𝑖𝑓 𝜃(𝐻𝑗) < 𝜃(𝐻𝑖)

𝐻𝑖 , 𝑖𝑓 𝜃(𝐻𝑖) = 𝜃(𝐻𝑗) 𝑎𝑛𝑑 |𝐻𝑖| < |𝐻𝑗|

𝐻𝑗 , 𝑖𝑓 𝜃(𝐻𝑖) = 𝜃(𝐻𝑗) 𝑎𝑛𝑑 |𝐻𝑗| < |𝐻𝑖|

{𝐻𝑖 , 𝐻𝑗} , 𝑖𝑓 𝜃(𝐻𝑖) = 𝜃(𝐻𝑗) 𝑎𝑛𝑑 |𝐻𝑖| = |𝐻𝑗|

In addition, we construct another commutative semihypergroup (𝐻, 𝑜) by considering

𝜆(𝐻𝑖) as the set of all minimal dominating sets which have minimum cardinality among all

minimal dominating sets that are obtained from 𝐻𝑖.

We also make a connection between minimal dominating sets and hyperstructures. let

𝑆 be the set of all minimal dominating sets and 𝑆𝑖 be a minimal dominating set. Then we

define 𝜑(𝑆𝑖) by

𝜑(𝑆𝑖) = 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑖𝑠𝑜𝑙𝑎𝑡𝑒𝑑 𝑣𝑒𝑟𝑡𝑖𝑐𝑒𝑠 𝑜𝑓 𝐺[𝑆𝑖] ∕ |𝑆𝑖|

We introduce the commutative semihypergroup (𝑆,∗𝑖) as follows:

for every 𝑆𝑚, 𝑆𝑛 ∈ 𝑆

𝑆𝑚 ∗𝑖 𝑆𝑛 = {

𝑆𝑛 , 𝑖𝑓 𝜑(𝑆𝑚) < 𝜑(𝑆𝑛) 𝑆𝑚 , 𝑖𝑓 𝜑(𝑆𝑚) > 𝜑(𝑆𝑛){𝑆𝑚, 𝑆𝑛} , 𝑖𝑓 𝜑(𝑆𝑚) = 𝜑(𝑆𝑛)

We investigate some situations in which this semihypergroup is hypergroup and give some

examples to clarify them. Finally, we present a new class of graphs in which this

semihypergroup will be a hypergroup.

Keyword(s): Dominating sets in graphs: Minimal dominating sets in graphs:

Semihypergroup: Hypergroup

2010 AMS Classification: 05C69, 20N20

31



1 Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran, Iran 2 Faculty of Sciences and Modern Technologies, Graduate University of Advanced Technology, Kerman, Iran


References:

1. Chartrand G., Lesnaik L., Graphs & Digraphs, Chapman & Hall, 1996.

2. Corsini P, Leoreanu V., Applications of Hyperstructure Theory, Kluwer Academic

Publishers, 2003.

3. Kalampakas A., Spartalis S., Path hypergroupoids: Commutativity and graph connectivity,

European Journal of Combinatorics, 44, 257–264, 2015.

4. Kalampakas A., Spartalis S., Tsigkas A., The Path Hyperoperation, Analele Stiintifice ale

Universitatii Ovidius Constanta, Seria Matematica, 22, 141-15, 2014.

5. Rosenberg I., Hypergroups induced by paths of a directed graph, Italian Journal of Pure

and Applied Mathematics, 4, 133-142, 1998.

32



1 Department of Mathematics, University of Bojnord, Bojnord, Iran

[email protected], [email protected], [email protected]

Ideals in HvMV-algebras

Mahmood Bakhshi 1 , RoghayehTaherpoor 1 and Akbar Paad 1

In this paper, first some basic definitions are reviewed. Then some types of ideals such

as Hv MV-ideals, weak Hv MV-ideals and nodal Hv MV-ideals are introduced and some

characterizations and their properties are obtained.

Introduction

In 1958, Chang [3], introduced the concept of an MV-algebra as an algebraic proof of

the completeness theorem for ℵ0-valued Łukasiewicz propositional calculus. After that many

mathematicians haveworked on MV-algebras and obtained significant results. Mundici [7]

proved that MV-algebras and AbelianA-groups with strong unit are categoricallyequivalent.

After that Marty[6]introduced the notion of a hypergroup several authors worked on

hypergroups, especially in France and in the United States, but also in Italy, Russia and Japan.

Bakhshi et al. introduced ordered polygroups [2] which are subclasses of hypergroups on

which is defined a partial ordering with special property. Hyperstructureshave many

applications to several sectors of both pure and applied sciences. A short review of the theory

of hyperstructures appear in [4]. Vougiouklis [8] introduced a generalization of the well-

known algebraic hyperstructures such as hypergroup so-called Hv-structures. Actually some

axioms concerning the above hyperstructures such as the associative law, the distributive law

and so on are replaced by their corresponding weak axioms. In order to obtain a suitable

generalization of MV-algebras which may be equivalent (categorically) to a certain subclass

of the class of Hv -groups, the author introduced the concept of an Hv MV-algebra [1] and

obtained some related results.

HvMV-algebras: Basic properties

In this section, the concept of an HvMV-algebra is introduced. For more details we

refer to thereferences.

Deftnition 2.1.An Hv MV-algebra is a nonempty set H endowed with a binary hyperoperation

‘⊕’, a unary operation ‘∗’ and a constant ‘0’ satisfying the followingconditions:

(HvMV1) x⊕(y⊕z)∩(x⊕y)⊕z≠ ∅, (weakassociativity)

(HvMV2) x⊕y∩y⊕x≠ ∅, (weakcommutativity)

(Hv MV3) (x∗)∗=x,

33





(Hv MV4) (x∗ ⊕ y)∗ ⊕ y ∩ (y∗ ⊕ x)∗⊕ x ≠∅,

(HvMV5) 0∗∈x⊕0∗∩0∗⊕x,

(Hv MV6) 0∗ ∈ x ⊕ x∗ ∩ x∗ ⊕ x, (Hv MV7) x∈ x ⊕ 0∩ 0 ⊕ x,

(Hv MV8) 0∗ ∈ x∗ ⊕ y ∩ y ⊕ x∗ and 0∗ ∈ y∗ ⊕ x ∩ x ⊕ y∗ imply x = y.

Remark 2.2. On any Hv MV-algebra H, a binary relation ‘ ≼ ’ by

x ≼ y⇔ 0∗ ∈ x∗ ⊕ y ∩ y ⊕ x∗

is introduced which is reflexive and antisymmetric but not necessarily transitive.

HvMV-ideals

In this section, the ideal theory of HvMV-algebras is studied. The concepts of weak HvMV-

ideal and HvMV-ideal are introduced and some properties and fundamental results are

obtained.

Deftnition 3.1.Let I be a nonempty subset of H satisfying (𝐼0) x ≼ y and y ∈ I imply x ∈ I.

Then, I is called

1. an HvMV-ideal if x⊕y⊆I,for all x,y∈I,

2. a weak HvMV-idealif x⊕y≼ I, for all x,y ∈I.

Theorem 3.2. Every Hv MV-ideal is a weak Hv MV-ideal.

Theorem 3.4. A non empty subset I of H is a weak HvMV-ideal if and only if it satisfies (𝐼0)

and (x⊕y)∩I≠ ∅ , for all x,y∈I.

Theorem 3.7. If {Iα :α ∈ Λ} is a nonempty family of Hv MV-ideals of H, then ⋂ 𝐼𝛼∈Λ 𝛼 is a

HvMV-ideal.

Deftnition 3.8. Let A be a nonempty subset of H and {Iα : α ∈ Λ} be a family of Hv

MV-ideals of H containing A. Then ⋂ 𝐼𝛼∈Λ 𝛼 is called the Hv MV-ideal generated by A,

denoted by (A).

Remark 3.12. It seems that Theorem 3.7 does not hold for weak Hv MV- ideals, ingeneral. At

this point, we can’t find any example showing this(open problem). But, there is a situation in

which we know that the intersection of a family of weak Hv MV-ideals is again a weak Hv

MV-ideal (see Theorem 3.17). When the same result holds for weak Hv MV-ideals, the weak

Hv MV-ideal generated by a family A of weak Hv MV-ideals of H is denoted by (𝐴)𝑤 .

Let Hv MVI (WHv MVI ) denotes the set of all Hv MV-ideals (weak Hv MV- ideals) of H.

Then, HvMVI (WHv MVI ) together with the set inclusion, as a partial ordering, is a poset.

34





Theorem3.13.(HvMVI,⊆)is a complete lattice and if WHvMVI isclosed with respect to the

intersection, HvMVI is a complete sublattice of the complete lattice (WHv MVI,⊆).

Deftnition 3.14. An element a ∈ H is called a right scalar if for all x ∈ H, |x ⊕ a| = 1, i.e., the

set x ⊕ a is singleton. We denote the set of all right scalars of H by R(H).

Deftnition 3.16.A subset S of H is said to be ⊕-closed, if for all x, y ∈ S, x ⊕ y ⊆S.

Theorem 3.17. Assume that |x ⊕ y| <∞, for all x, y ∈ H, ≼ is transitive and monotone and

R(H) is ⊕-closed. If A is a nonempty subset of H contained in R(H), then

(𝐴)𝑤={x∈H: x ≼ (•••((𝑎1⊕𝑎2)⊕•••)⊕ 𝑎𝑛, for some n∈N, 𝑎1,..., 𝑎𝑛∈A}.

Particularly, if A = {a}, (𝑎)𝑤= {x ∈ H : x na, for some n ∈ N}.

Nodal HvMV-ideals

Deftnition 4.1. Let H be an HvMV-algebra. By a(weak)nodal HvMV-ideal of H we mean a

(weak) Hv MV-ideal of H which is comparable with each (weak) HvMV-ideal of H.

Example4.2.Consider the HvMV-algebra (H;⊕,∗,0), where ⊕ and ∗ are defined as in

Table4. Routine calculations show that the only HvMV-ideals of H are{0}, {0, a} and H. So,

every Hv MV-ideal of H is a nodal Hv MV-ideal.

⊕ 0 a b c 1

0 {0} {0,a} {0, b} {0,c} {0, a, b, c, 1}

a {0,a} {0,a} {0, a, b, c, 1} {0, a, b, c, 1} {0, a, b, c,1}

b {0, b} {0, a, b, c, 1} {0, a, b, c, 1} {0, a, b, c} {0, a, b, c,1}

c {0,c} {0, a, b, c, 1} {0, a, b, c} {0, a, b, c, 1} {0, a, b, c,1}

1 {0,a,b,c,1} {0,a,b,c,1} {0,a,b,c,1} {0,a,b,c,1} {0,a,b,c,1}

∗ 1 b a c 0

Table 4: Cayley table of Hv MV-algebra given in Example4.2

Proposition 4.5. Any nodal Hv MV-ideal is a nodal weak Hv MV-ideal.

Theorem 4.6. LetI be a(weak)HvMV-ideal of H. If for every x∈I and for every y∈H\I, x≼y,

then I is a nodal(weak)HvMV-ideal.

35





Theorem 4.7. In an HvMV-algebra with a totally ordered, any(weak)HvMV- ideal is a

nodal(weak)HvMV-ideal.

Theorem4.8. Assume that the conditions of Theorem3.17 holds for HvMV- algebra H. If x ∈

R(H) is a node, (𝑥)𝑤 is also a nodal weak Hv MV-ideal of H.

For weak Hv MV-ideal I of H and x ∈ H, the weak Hv MV-ideal of H Generated by I∪{x}

will denoted by I(x).

Theorem 4.10. Assume that the conditions of Theorem3.17 holds for HvMV- algebra H. If I

is a nodal weak Hv MV-ideal of H and x ∈ R(H) is a node, then I(x) is a nodal weak Hv MV-

ideal of H.

Theorem4.11. The intersection of any nonempty family of nodal HvMV- ideals is again a

nodalHvMV-ideal.

Considering Theorem3.13 we get

Corollary 4.12. Let N (H) be the set of all nodal Hv MV-ideals of H. Then (N (H), ⊆) is a

complete sublattice of (Hv MVI ,⊆).

Keyword(s): hyperstructures, algebras of logics.


Reference(s):

1. Bakhshi M., Hv MV-algebras I, Quasigroups Related Systems,22, 9-18, 2014.

2. Bakhshi M. and Borzooei R.A.,Ordered polygroups, Ratio Mathematica, 24,31-40, 2013.

3. Chang C. C., Algebraic analysis of many valued logics, Trans. Amer. Math. Soc. 88,

467-490, 1958.

4. Corsini P. and Leoreanu V., Applications of Hyperstructure Theory, Kluwer Academic

Publishers, Dordrecht, 2003.

5. Ghorbani Sh., Hassankhani A. and Eslami E., Hyper MV-algebras, Set-Valued

Mathematics and Applications, 1, 205-222, 2008.

6. Marty F., Sur une generalization de la notion de groups, 8th congress Math. Scandinaves,

Stockhholm, 45-49, 1934.

7. Mundici D., Interpretation of AFC∗-algebras in Łukasiewiczsententialcalsulus, J.

Func.Anal., 65, 15-63, 1986.

8. Vougiouklis T., A new class of hyperstructures, J. Combin. inform. Syst. Sci. 20, 229–235,

1995.

36




2Centre for Systems and Information Technologies, University of Nova Gorica, Vipavska cesta 13, 5000 Nova Gorica, Slovenia,

3Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technická 8, 616 00 Brno, Czech Republic,

E-mails: [email protected]

Overview on the Height of a Hyperideal in Krasner

Hyperrings

Hashem Bordbar1, Irina Cristea

2, and Michal Novak

3

Similarly as in ring theory, the notion of height of a prime hyperideal of a hyperring

has recently been defined and studied [1], extending the concept of dimension of a hyperring,

in the context of commutative Krasner hyperrings. These are hyperstructures endowed with an

additive hyperoperation and a multiplicative operation, satisfying certain properties,

introduced by Krasner [7] as a tool in the approximation of valued fields. The height of a

proper prime hyperideal of a Krasner hyperring is defined as the maximum of the lengths of

the chains of distinct prime hyperideals contained in it, or it is ∞ if such a number does not

exist.

One of the most significant theorems in commutative algebra is called Krull’s height

theorem or Krull’s principal ideal theorem. Using the properties of prime hyperideals in

Noetherian Krasner hyperrings, we present an extension of this theorem [1]: If R is a

commutative Krasner hyperring and I is a proper principal hyperideal of R, then the height of

a minimal prime hyperideal of R over I is at most 1. Later on in [2], we extended this result,

proving that in a commutative Krasner hyperring R, the height of a minimal prime hyperideal

over a proper hyperideal of R generated by n elements is at most n. The converse of this

theorem is also true.

Our future goal is to extend these results to other classes of hyperrings, highlighting

their differences/similarities with the classical results for commutative rings.

Keywords: Krasner hyperring, prime/maximal hyperideal, Noetherian hyperring,

height of a prime hyperideal, dimension of a hyperring


Reference(s):

1. H. Bordbar, I. Cristea, Height of prime hyperideals in Krasner hyperrings, Filomat

(accepted for pubblication in 2017).

2. H. Bordbar, I. Cristea, M. Novak, Height of hyperideals in Noetherian Krasner hyperrings,

U.P.B. Sci. Bull., Series A, 79(2017), no. 2, 31-42.

3. I. Cristea, S. Jancic-Rašovic, Composition hyperrings, An. St. Univ. Ovidius Constanta,

21(2013), no.2, 81-94.

37




2Centre for Systems and Information Technologies, University of Nova Gorica, Vipavska cesta 13, 5000 Nova Gorica, Slovenia,

3Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technická 8, 616 00 Brno, Czech Republic,

E-mails: [email protected]

4. B. Davvaz, V. Leoreanu-Fotea, Hyperring theory and applications, International

Accademic Press, Palm Harbor, U.S.A., 2007.

5. D. Eisenbud, Commutative algebra, Graduate Texts in Mathematics 150, Berlin, New

York: Springer-Verlag, 1995.

6. I. Kaplansky, Commutative rings, University of Chicago Press, Chicago, 1974.

7. M. Krasner, Approximation des corps values complets de caracteristique p; p > 0, par ceux

de caracteristique zero, Colloque d’Algebre Superieure (Bruxelles, Decembre 1956), CBRM,

Bruxelles, 1957.

8. Ch.G. Massouros, On the theory of hyperrings and hyperfields, Algebra i Logika, 24(1985),

728-742.

9. J. D. Mittas, Hyperanneaux canoniques, Math. Balkanica, 2(1972), 165- 179.

10. A. Nakassis, Recent results in hyperring and hyperfield theory, Int. J. Math. Math. Sci.,

11(1988), 209-220.

11. S. Spartalis, A class of hyperrings, Riv. Mat. Pura Appl. 4 (1989), 55-64.

12. T. Vougiouklis, The fundamental relation in hyperrings. The general hyperfields,

Algebraic hyperstructures and applications (Xanthi, 1990), 203–211, World Sci. Publ.,

Teaneck, NJ, 1991.

38




2Department of Mathematics Education, Gyeongsang National University, Jinju 52828, Korea

E-mail(s): [email protected]

Theory of Double-framed soft set theory on Hyper

BCK-algebra

Hashem Bordbar1 and Young Bae Jun

2

The notion of double-framed soft (strong) hyper BCK-ideal of hyper BCK-algebra is

introduced, and related properties are investigated. Characterization of double-framed soft

(strong) hyper BCK-ideal is considered, and relation between double-framed soft hyper BCK-

ideal and double-framed soft strong hyper BCK-ideal is discussed.

Keywords: Cubic intuitionistic set, cubic intuitionistic ideal, positive implicative

cubic intuitionistic ideal.

2010 AMS Classification: 06F35, 03G25, 06D72.

References:

1. Y. B. Jun and S. S. Ahn, Double-framed soft sets with applications in BCK/BCI-algebras,

J. Appl. Math. Volume 2012, Article ID 178159, 15 pages.

2. Y. B. Jun, G. Muhiuddin and A. M. Al-roqi, Ideal theory of BCK=BCI- algebras based on

double-framed soft sets, Appl. Math. Inf. Sci. 7(5), (2013), 1879–1887.

3. Y. B. Jun and C. H. Park, Applications of soft sets in ideal theory of BCK/BCI-algebras,

Inform. Sci. 178, (2008), 2466–2475.

4. Y. B. Jun and X. L. Xin, Scalar elements and hyperatoms of Hyper BCK-algebras,

Scientiae Mathematicae 2(3), (1999), 303–309.

5. Y. B. Jun, X. L. Xin, E. H. Roh and M. M. Zadehi, Strong hyperBCK- ideals of

hyperBCK-algerbas, Math. Japon. 51(3), (2000), 493–498.

39



1Democritus University of Thrace, School of Education, Alexandroupolis, Greece

2 Democritus University of Thrace, School of Education, Alexandroupolis, Greece


On P-hopes and P-Hv-structures on the plane

Achilles Dramalidis1, Ioanna Iliou

2

In this paper we deal with P-hyperstructures which are defined in Hv-

groups. Using a weak commutative hyperoperation on the plane and a

specific subset of this plane we construct various P-Hv-structures. In addition,

we study the existence of units and inverses of these constructions,

connecting them with Join Spaces, as well.

Keyword(s): Hyperstructures, Hv-structures, hopes, P-hyperstructures.


References:

1. Antampoufis N., Vougiouklis T., Hyperoperations greater than the Complex Number

operations, Journal of Basic Science, 3, 11-17, 2006.

2. Corsini P., Leoreanu V., Application of Hyperstructure Theory, Klower Ac.

Publ., 2003.

3. Davvaz B., Santilli R.M., Vougiouklis T., Studies of multivalued hyper-

structures for the characterization of matter-antimatter systems and their

extension, Algebras, Groups and Geometries 28, 105–116, 2011.

4. Dramalidis A., Some geometrical P-HV-structures, New Frontiers in

Hyperstructures, 93-102, 1996.

5. Dramalidis A., Dual Hv-rings, Rivista di Matematica Pura ed Applicata, 17,

55-62, 1996.

6. Dramalidis A., Vougiouklis T., Fuzzy Hv-substructures in a two dimensional

Euclidean vector space, Iranian J. Fuzzy Systems, 6, 1-9, 2009.

7. Iranmanesh A., Iradmusa M.N., Hv-structures associated with generalized P-

hyperoperations, Bul. of the Iranian Math. Soc., 24, 33–45, 1998.

8. Vougiouklis T., Generalization of P-hypergroups, Rendiconti Circolo

Matematico di Palermo, 36, 114-121, 1987.

9. Vougiouklis T., The fundamental relation in hyperrings. The general

hyperfield, World Scientific, 203-211, 1991.

10. Vougiouklis T., Hyperstructures and their Representations, Monographs

Math., Hadronic Press, 1994.

11. Vougiouklis T., Representations of hypergroups by generalized permutations,

Algebra Universalis, 29, 172-183, 1992.

12. Vougiouklis T., Dramalidis A., Hv-modulus with external P-hyperoperations,

Proc. of the 5th AHA, Iasi, Romania, 191-197, 1993.

40



1Kocaeli University, Department of Mathematics, Kocaeli, Turkey

2 Kocaeli University, Department of Mathematics, Kocaeli, Turkey

3 Kocaeli University, Department of Mathematics, Kocaeli, Turkey

E-mails:[email protected], [email protected], [email protected]

On Neutrosophic Linear Spaces

Banu Pazar Varol 1, Vildan Çetkin

2, Halis Aygün

3

Smarandache introduced the neutrosophy which is a branch of philosophy. Then Wang

et.al. defined single valued neutrosophic sets. Neutrosophic set is a part of neutrosophy which

studies the origin, nature and scope of neutralities. In neutrosophic set, truth-membership,

indeterminacy membership and false-membership functional values are independent. Single

valued neutrosophic set is applied to algebraic and topological structures. In this paper, we

introduce neutrosophic linear space over the neutrosophic field and consider its main

properties.

Keywords: Neutrosophic set, single valued neutrosophic set, linear space

2010 AMS Classification: 08A72, 06D72

References:

1. Nanda S., Fuzzy fileds and fuzzy linear spaces, Fuzzy sets and Systems, 19, 89-94, 1986.

2. Smarandache F., A unifying field in logics. Neutrosophy/ Neutrosophic Probability, Set

and Logic, Rehoboth: American Research Press (1998) http://fs.gallup.unm.edu/eBook-

neutrosophics6.pdf (last edition online).

3. Wang H. et al., Single valued neutrosophic sets, Proc. of 10th Int. Conf. on Fuzzy Theory

and Technology, Salt Lake City, Utah, July, 21-26, 2005.

41




1Recep Tayyip Erdoğan University, Department of Mathematics, Rize, Turkey

E-mail:[email protected]

The Relationships between the Orders Induced by Implications and Uninorms

M. Nesibe Kesicioğlu1

In this paper, an order by means of implications on a bounded lattice possessing some

special properties is defined and some of its properties are discussed. By giving an order

based on uninorms on a bounded lattice, the relationships between such generated orders are

investigated.

Keywords: Implications, partial order, bounded lattice, law of importation

2010 AMS Classification: 03E72, 03B52

References:

1. Baczynski M., Jayaram B., Fuzzy implications, Studies in Fuzziness and Soft Computing,

vol. 231, Springer, Berlin, Heidelberg, 2008.

2. Birkhoff G., Lattice Theory, 3 rd edition, Providence, 1967.

3. Jayaram B., A new ordering based on fuzzy implications, Proceedings of ISAS 2016,

Luxembourg, Luxembourg, 49-50, 2016.

4. Karaçal F., Kesicioğlu M.N., A T-partial order obtained from t-norms, Kybernetika, 47,

300-314, 2011.

5. Kesicioğlu M.N., Mesiar R., Ordering based on implications, Information Sciences, 276,

377-386, 2014.

6. Mas M., Monserrat M., Torrens J., A characterization of (U,N), RU, QL and D-

implications derived from uninorms satisfying the law of importation, Fuzzy Sets and

Systems, 161, 1369-1387, 2010.

42



1Recep Tayyip Erdoğan University, Department of Mathematics, Rize, Turkey 2Karadeniz Technical University, Department of Mathematics, Trabzon, Turkey 3Karadeniz Technical University, Department of Mathematics, Trabzon, Turkey

E-mails: [email protected], [email protected], [email protected]

A Survey on Order-equivalent Uninorms

M. Nesibe Kesicioğlu1, Ümit Ertugrul

2, Funda Karaçal

3

In this paper, an equivalence on the class of uninorms on a bounded lattice 𝐿 based on

the equality of the orders is discussed. Some relationships between the orders induced by t-

norms and their N-dual t-conorms are determined. Also, defining the set of all incomparable

elements w.r.t. the order induced by uninorms, some relationships with the sets of all

incomparable elements w.r.t. the orders induced by the corresponding underlying t-norm and

t-conorm are presented.

Keywords: Uninorm, bounded lattice, partial order, equivalence of uninorms

2010 AMS Classification: 03E72, 03B52

References:

1. Birkhoff G., Lattice Theory, 3 rd edition, Providence, 1967.

2. Ertuğrul Ü, Kesicioğlu M.N., Karaçal F., Ordering based on uninorms, Information

Sciences, 330, 315-327, 2016.

3. Grabisch M., Marichal J.-L., Mesiar R., Pap E., Aggregation Functions, Cambridge

University Press, 2009.

4. Karaçal F., Kesicioğlu M.N., A T-partial order obtained from t-norms, Kybernetika, 47,

300-314, 2011.

5. Kesicioğlu M.N., Karaçal F., Mesiar R., Order-equivalent triangular norms, Fuzzy Sets and

Systems, 268, 59-71, 2015.

6. Kesicioğlu M.N., Mesiar R., Ordering based on implications, Information Sciences, 276,

377-386, 2014.

7. Klement E.P., Mesiar R., Pap E., Triangular Norms, Kluwer Academic Publishers,

Dordrecht, 2000.

43



1 State University of Moldova 2State University from Tiraspol

E-mails: [email protected], [email protected]

THE REFLECTOR FUNCTOR AND THE LATTICE 𝕃(𝓡)

Cerbu Olga1 and Dumitru Botnaru

1

In the category 𝒞2𝒱 of locally convex topological vector spaces [RR] we examine a

class of factorization structures for which the reflector functor transforms the class of

projections or the class of injections, or both classes into themselves. Such functors are

usually studied (see [K], [G], [B], [BC], [C]) and appear when studying the semireflexive

subcategories [CB].

Let 𝛱 be the subcategory of the complete spaces with the weak topology and

𝜋: 𝒞2𝒱 → 𝛱 - the reflector functor. The subcategory 𝛱 is the minimal element in the lattice ℝ.

Let ℛ ∈ ℝ. For every object 𝑋 of the category 𝒞2𝒱 let be 𝑟𝑋: 𝑋 → 𝑟𝑋 and 𝜋𝑋: 𝑋 → 𝜋𝑋 the ℛ

and 𝛱-repliques. Since 𝛱 ⊂ ℛ, we have

𝜋𝑋 = 𝑣𝑋𝑟𝑋

for a morphism 𝑣𝑋 . We denote 𝒰 = 𝒰(ℛ) = {𝑟𝑋 ∣ 𝑋 ∈∣ 𝒞2𝒱 ∣}, 𝒱 = 𝒱(ℛ) = {𝑣𝑋 ∣

𝑋 ∈∣ 𝒞2𝒱 ∣}. We have the following factorization structures:

, , , , , ., ,P P P P └ └ └

For ℛ ∈ ℝ we denote by 𝕃(ℛ) the class of factorization structures (ℰ,ℳ), for which

𝒫ʹ(ℛ) ⊂ ℰ ⊂ 𝒫ʺ(ℛ) and 𝕃𝑢(ℛ) = {(ℰ,ℳ) ∈ 𝕃(ℛ) ∣ ℳ ⊂ℳ𝑢} (see [B]), where ℳ𝑢 is a

class of the universal monomorphisms (see [B]).

.,p uu u uP P

└

Definition 1. Let 𝑟: 𝒞 → ℛ be a covariant functor, and (𝒫, ℐ) - a factorization structure (the

left or right factorization structure). We say that this functor 𝑟 is:

1. P-functor, if r(P) P.

2. I-functor, if r(I) I.

3. (P; I)-functor, if r(P) P and r(I) I.

Proposition 2. 1. 𝕃𝑢(ℛ) is a complete lattice with the minimal element (𝒫ʹ𝑢, ℐʹ𝑢) and the

maximal element (𝒫ʺ, ℐʺ).

2. 𝕃𝑢(ℛ) is the class of the factorization structures (ℰ,ℳ), for which ℐʹ𝑢 ⊂ ℳ ⊂ ℐʺ.

Lemma 3. Let 𝑚:𝑋 → 𝑌 be an universal monomorphism. Then 𝜋(𝑚) is a sectional

morphism.

Theorem 4. 1. Let (ℰ,ℳ) ∈ 𝕃𝑢(ℛ). Then 𝑟: 𝒞2𝒱 → ℛ is a (ℰ,ℳ)-functor: 𝑟(ℰ) ⊂ ℰ and

𝑟(ℳ) ⊂ ℳ.

2. 𝑓 ∈ 𝒫ʺ(ℛ) ⇔ 𝑟(𝑓) ∈ 𝒫ʺ(ℛ).

44



1 State University of Moldova 2State University from Tiraspol


Keywords: Reflector functor, lattice

2010 AMS Classification: 55P65

References:

1. [B] Botnaru D., Structures bicatégorielles complémentaires. ROMAI J., v.5, Nr.2, 2009,

p.5-27.

2. [BC] Botnaru D., Cerbu O., Functor of Special Type, Proceedings of the VIII International

Workshop, Lie Theory and its Applications in Physics, Bulgaria Academy of Sciences,

Institute for Nuclear Reasearch and Nuclear Energy, Varna, Bulgaria, 15-21 June 2009, p.

299-311

3. [CB] Cerbu O., Botnaru D., Some properties of semireflexivity, Noncommutative structures

in mathematics and physics, 22-26 July 2008, Brussels, p.71-84.

4. [C] Cerbu O., The Lattice of Semireflexive Subcategories, ROMAI Journal, CAIM,

Universitatea Al. Ioan Cuza, Septembrie 2010, Iaşi

5. [G] Grothendieck A., Topological vector spaces, Gordon and Breach, 1973.

6. [K] Kennison J. F., Reflective functors in general topology, TAMS, 1965, v.118, p.309-

315.

7. [RR] Robertson A. P., Robertson W. J., Topological vector spaces, Cambridge, England,

1964.

45



1Yildiz Technical University, Department of Mathematics, Istanbul, Turkey

E-mails:,[email protected], [email protected], [email protected], [email protected]

2-Absorbing -Primary Fuzzy Ideals of Commutative Rings

Deniz Sönmez1, Gürsel Yeşilot

1, Serkan Onar

1 and Bayram Ali Ersoy

1

In this work, we define 2-Absorbing -primary fuzzy ideals which is the

generalizations of 2-absorbing fuzzy ideal and 2-absorbing primary fuzzy ideals. Furthermore,

we give some fundamental results concerning these notions.

Keywords: 2-Absorbing -Primary fuzzy ideals, 2-Absorbing primary fuzzy ideals.

2010 AMS Classification: 03E72

References:

1.Zadeh L.A., Fuzzy sets, Inform. and Control, 8, 338-353, 1965.

2.A. Badawi , On 2-absorbing ideals of commutative rings, Bull. Austral. Math. Soc. 75 (2007), no. 3,

417-429.

3. A. Badawi, U. Tekir, E. Yetkin, On 2-absorbing primary ideals in commutative rings, Bull. Austral.

Math. Soc. 51 (2014), no. 4, 1163-1173.

4. F. Callialp, E. Yetkin and U. Tekir, On 2-absorbing primary and weakly 2-absorbing primary

elements in multiplicative lattices, Italian Journal of Pure and Applied Mathematics, 34 (2015), 263-

276 .

5. Badawi A. and Darani A.Y, On weakly 2-absorbing ideals of commutative rings. Houston J. Math.

39, 441-452, 2013.

6. Malik D.S. and Mordeson J.N., Fuzzy Commutative Algebra, World Scienti_c Publishing, 1998.

7. V.N. Dixit, R. Kumar and N. Ajmal. Fuzzy ideals and fuzzy prime ideals of a ring, Fuzzy Sets and

Systems 44 (1991), 127-138.

8.W.J. Liu, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets and Systems 8 (1982), 133-139.

9.T.K. Mukherjee and M.K. Sen, Prime fuzzy ideals in rings, Fuzzy Sets and Systems 32 (1989), 337-

341.

10. T.K. Mukherjee and M.K. Sen, Primary fuzzy ideals and radical of fuzzy ideals, Fuzzy Sets and

Systems 56 (1993), 97-101.

11.L.I. Sidky , S.A. Khatab, Nil radical of fuzzy ideal, Fuzzy Sets and Systems 47 (1992), 117-120.

46



1 Department of Mathematics, Shahid Beheshti University, Tehran, Iran

2Department of Mathematics, Alzahra University, Tabnak Street, Vanak Square,

E-mails: [email protected] , [email protected]

Relation Between Hyper EQ-algebras and Some Other Hyper Structures

Rajab Ali Borzooei1 and Batol Ganji Saffar

2

In this study by considering the notion of hyper EQ-algebra, as a generalization

of EQ-algebra (algebra of truth values for a higher-order fuzzy logic), we define some types

of filters in this structuer and investigated some related results. Then we find the relation

between hyper EQ-algebras and hyper BCK-algebras, hyper MV -algebras and (weak) hyper

residuated lattices.

Keywords: Hyper EQ-algebra, hyper BCK-algebra, hyper MV -algebra, (weak) hyper

residuated lattice


References:

47



1,2 Karadeniz Technical University, Department of Mathematics, Trabzon, Turkey 3 Yazd University, Department of Mathematics, Yazd, Iran

E-mail(s): [email protected] , [email protected] , [email protected]

Fuzzy hyperideals in ordered semihyperrings

Osman KAZANCI1, Şerife YILMAZ

2, Bijan DAVVAZ

3

In this study, we introduce the concept of fuzzy hyperideals of ordered

semihyperrings, which is a generalization of the concept of fuzzy hyperideals of

semihyperrings to ordered semihyperring theory. We investigate its related properties. We

show that every fuzzy quasi-hyperideal is a fuzzy bi-hyperideal and in a regular ordered

semihyperring, fuzzy quasi-hyperideal and fuzzy bi-hyperideal coincide.

Keywords: Semihyperring: ordered semihyperring: fuzzy hyperideal.

2010 AMS Classification: 03E72; 97H50.

Reference(s):

1. Ameri R., Hedayati H., On k-hyperideals of semihyperrings, J. Discrete Math. Sci.

Cryptogr., 10(1), 41-54, 2007.

2. Corsini P., Leoreanu-Fotea V., Applications of Hyperstructure Theory, Kluwer Academic

Publishers, Dordrecht, The Netherlands, 2003.

3. Davvaz B., Cristea I., Fuzzy Algebraic Hyperstructures, An Introduction, Springer, 2015.

4. Davvaz B., Leoreanu-Fotea V., Fuzzy ordered Krasner hyperrings, Journal of Intelligent

and Fuzzy Systems, 29, 2015.

5. Heidari D., Davvaz B., On ordered hyperstructures, Politech. Univ. Bucharest Sci. Bull.

Ser. A. Appl. Math. Phys., 73(2), 85-96, 2011.

6. Kazancı O., Yamak S., Generalized fuzzy bi-ideals of semigroup, Soft Computing, 12,

1119-1124, 2009.

7. Kehayopulu N., Tsingelis M., Fuzzy Right, Left, Quasi-Ideals, Bi-Ideals in Ordered

Semigroups, Lobachevskii Journal of Mathematics, 30, 17-22, 2009.

8. Rosenfeld A., Fuzzy groups, J. Math. Anal. Appl. 35, 512-517, 1971.

9. Vougiouklis T., Hyperstructures and their Representations, Hadronic Press, Florida, USA,

1994.


48






1,2 Karadeniz Technical University, Department of Mathematics, Trabzon, Turkey

E-mail(s): [email protected] , [email protected]

Fuzzy interior hyperideals in ordered semihyperrings

Şerife YILMAZ 1, Osman KAZANCI

2

In this study, we introduce the concept of fuzzy interior hyperideals in ordered

semihyperrings, which are a new sort of fuzzy hyperideals of semihyperrings. We investigate

some of their related properties. We give a characterization of fuzzy interior hyperideals in

terms of their level subsets. We show that every fuzzy hyperideal is a fuzzy interior

hyperideal. We introduce the concept of intra-regular ordered semihyperrings and show that

fuzzy hyperideals and fuzzy interior hyperideals coincide in an intra-regular ordered

semihyperring. Finally, we introduce the concept of fuzzy simple ordered semihyperrings and

prove some results.

Keywords: Ordered semihyperring: interior hyperideal: fuzzy interior hyperideal.

2010 AMS Classification: 03E72; 97H50.

Reference(s):

1. Ameri R., Hedayati H., On k-hyperideals of semihyperrings, J. Discrete Math. Sci.

Cryptogr., 10(1), 41-54, 2007.

2. Corsini P., Leoreanu-Fotea V., Applications of Hyperstructure Theory, Kluwer Academic

Publishers, Dordrecht, The Netherlands, 2003.

3. Davvaz B., Cristea I., Fuzzy Algebraic Hyperstructures, An Introduction, Springer, 2015.

4. Hedayati G., t-implication-based fuzzy interior hyperideals of semihypergroups, Journal of

Discrete Mathematical Sciences and Cryptography, 13(2), 123-140, 2010.

5. Kazancı O., Yamak S., Generalized fuzzy bi-ideals of semigroup, Soft Computing, 12,

1119-1124, 2009.

6. Kehayopulu N., Tsingelis M., Fuzzy Right, Left, Quasi-Ideals, Bi-Ideals in Ordered

Semigroups, Lobachevskii Journal of Mathematics, 30, 17-22, 2009.

7. Rosenfeld A., Fuzzy groups, J. Math. Anal. Appl. 35, 512-517, 1971.

8. Tang J., Davvaz B., Luo Y., A study on fuzzy interior hyperideals in ordered

semihypergroups, Italian Journal of Pure and Applied Mathematics, 36, 125-146, 2016.


49





1Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Iran


HYPERHILBERT SPACES

SAEID GHOLAMPOOR1 and ALI TAGHAVI

1

In this study we introduce the concept of hyper Hilbert spaces and prove some result

such as, orthogonal projection and Riesz theorem about them.

Keywords: Hyperhilbert space, hilbert space, hyper space


References:

1. P. Corsini, Prolegomena of hypergroup theory, Aviani editore, (1993).

2. P. Corsini and V. Leoreanu, Applications of Hyperstructure theory, Kluwer Academic Publishers,

Advances in Mathematics (Dordrecht), (2003).

3. F. Marty, Sur nue generalizeation de la notion de group, 8th congress of the Scandinavic

Mathematics, Stockholm, (1934), 45{49.

4. S. Roy and T. K. Samanta, Innerproduct hyperspaces, Accepted in Italian J. of Pure and Appl. Math.

5. A. Taghavi and R. Hosseinzadeh, A note on dimension of weak hypervector spaces, Italian J. of

Pure and Appl. Math, To appear.

6. A. Taghavi and R. Hosseinzadeh, Hahn-Banach Theorem for functionals on hypervector spaces,

The Journal of Mathematics and Computer Science, Vol .2 No.4 (2011) 682-690.

7. A. Taghavi and R. Hosseinzadeh, Operators on normed hypervector spaces, Southeast Asian

Bulletin of Mathematics, (2011) 35: 367-372.

8. A. Taghavi and R. Hosseinzadeh, Operators on weak hypervector spaces, Ratio Mathematica, 22

(2012) 37-43.

9. A. Taghavi and R. Hosseinzadeh and H. Rohi, Hyperinner product spacess,

10. A. Taghavi and T. Vougiouklis and R. Hosseinzadeh, A note on Operators on Normed Finite

Dimensional Weak Hypervector Spaces, Scientic bulletin, Series A, Vol. 74, Iss. 4 (2012) 103-108.

11. M.Scafati-Tallini, Characterization of remarkable Hypervector space, Proc. 8th congress on

"Algebraic Hyperstructures and Aplications", Samotraki, Greece, (2002), Spanidis Press, Xanthi,

(2003), 231-237.

12. M.Scafati-Tallini, Weak Hypervector space and norms in such spaces, Algebraic Hyperstructures

and Applications Hadronic Press. (1994), 199-206.

13. T. Vougiouklis, The fundamental relation in hyperrings. The general hyper_eld. Algebraic

hyperstructures and applications (Xanthi, 1990), World Sci. Publishing, Teaneck, NJ, (1991),

203{211.

14. T. Vougiouklis, Hyperstructures and their representations, Hadronic Press, (1994).

15. M. M. Zahedi, A review on hyper k-algebras, Iranian Journal of Mathematical Sciences and

Informatics, Vol. 1, No. 1 (2006). 55-112.

50



1Namık Kemal Univesity, Department of Mathematics, Tekirdağ, Turkey.

2 Karadeniz Technical Univesity, Department of Mathematics, Trabzon, Turkey.

E-mail(s): [email protected], [email protected], [email protected].

The Lattice Structure of Subhypergroups of a Hypergroup

Dilek BAYRAK1, Sultan YAMAK

2, Şerife YILMAZ

2

In mathematics, determination of algebraic structures is very important. Many

methods have been applied in determining these structures until today. One of them is

investigating the lattice structure of substructures of algebraic structures (such as submonoids

of a monoid, subgroups of a group, ideals of a ring, submodules of a module, subspaces of a

vector space, etc.) according to the inclusion relation. As a generalization of algebraic

structures, hyper structure was defined in 1934 by F. Marty. Since then this theory has been

developed by many mathematicians. In the last fifteen years, various applications of algebraic

structures (in geometry, binary relations, lattices, fuzzy sets, rough sets, automata,

cryptography, codes, median algebra, relational algebra, artificial intelligence probability)

have been obtained.

In this study, we investigate the properties of closed, invertible, ultraclosed and

conjugable subhypergroups classes. We study when the hypergroups satisfy the property that

the hyperproduct of subhypergroups becomes an operation on the set of subhypergroups. It is

investigated in which cases, the poset of the subhypergroups of a hypergroup is a lattice. It is

examined when this lattice is modular or distributive. Thus some information about a

hypergroup may be obtained by investigating the lattice of its subhypergroups.

Keyword(s): subhypergroups, lattice.

2010 AMS Classification: 20N20, 06B99.

Reference(s):

1. Birhoff G., Lattice Theory, Amer. Math. Soc. Colloq. Publ., 1967.

2. Corsini P. and Leoreanu V., Application of Hyperstructure Theory, Kluwer Academic

Publishers, 2003.

3. Davvaz B., Leoreanu V., Hyperring theory and applications, International Academic Press,

2007.

4. Marty, F., Sur ungeneralisation de la notion degroup, 8th Congress of Scandinavian

Mathematicians, 45-49, 1934.

5. Massouros, C.G., Some properties of certain subhypergroups, Ratio Mathematica, 25, 67-

76, 2013.

6. Schmidt R., Subgroup Lattices of Groups, de Gruyter Expositions in Mathematics 14, de

Gruyter, Berlin, 1994.

7. Tarnauceanu, M., On the poset of subhypergroups of a hypergroup, Int. J. Open Problems

Comp. Math. 3(2), 115-122, 2010.

51




Emails: [email protected], [email protected], [email protected], [email protected]

Intuitionistic Fuzzy Weakly Prime Ideals

Tuğba Arkan1, Serkan Onar

1, Deniz Sönmez

1, Bayram Ali Ersoy

1

In this study, the fundamental definitions and theorems regarding intuitionistic fuzzy

sets and intuitionistic fuzzy ideals of commutative ring with identity 𝑅 have been given as

preliminaries. After the preliminaries, we introduce the notions of intuitionistic fuzzy weakly

prime ideals, intuitionistic fuzzy partial weakly prime ideals, intuitionistic fuzzy weakly

semiprime ideals of 𝑅. Let 𝑃 = ⟨𝜇𝑃, 𝜐𝑃 ⟩ be a nonconstant intuitionistic fuzzy ideal of 𝑅. If

(0,1) ≠ 𝐴 ∙ 𝐵 ⊆ P implies 𝐴 ⊆ 𝑃 or 𝐴 ⊆ 𝑃 where 𝐴 = ⟨𝜇𝐴, 𝜐𝐴 ⟩ , 𝐵 = ⟨𝜇𝐵, 𝜐𝐵 ⟩ intuitionistic

fuzzy ideals of 𝑅, then 𝑃 is called intuitionistic fuzzy weakly prime ideal of 𝑅. If 𝑃(𝑥𝑦) =

𝑃(𝑥) or 𝑃(𝑥𝑦) = 𝑃(𝑦) for 𝑥𝑦 ≠ 0, then 𝑃 is called intuitionistic fuzzy partial weakly prime

ideal of 𝑅. A nonconstant intuitionistic fuzzy ideal 𝑃 is called intuitionistic fuzzy weakly

semiprime ideal of 𝑅 if (0,1) ≠ 𝐵2 ⊆ 𝑃 implies 𝐵 ⊆ 𝑃 where 𝐵 is an intuitionistic fuzzy

ideal of 𝑅. Also, we give some relations between intuitionistic fuzzy weakly prime ideals and

weakly prime ideals of 𝑅.

Keywords: Intuitionistic fuzzy prime ideals, intuitionistic fuzzy weakly prime ideals,

intuitionistic fuzzy partial weakly prime ideals, intuitionistic fuzzy weakly semiprime ideals.

2010 AMS Classification: 03F55, 03E72, 08A72.

References:

1. Atanassov K. , Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20, 87–96, 1986.

2. Hur, K., Jang, S. Y. & Kang, H. W., Intuitionistic fuzzy ideal of a ring, J. Korea Soc. Math.

Educ. Ser. B: Pur Appl. Math., Vol 12, 2005.

3. P. A. Ejegwa, A. J. Akubo, O. M. Joshua, Intuitionistic Fuzzy Set and Its Application in

Career Determination via Normalized Euclidean Distance Method, European Scientific

Journal edition vol.10, No.15, 2014.

4. D. D. Anderson and E. Smith, Weakly prime ideals, Houston J. Math., 29, 4, 831–840,

2003.

5. J.N Mordeson, D.S. Malik, Fuzzy Commutative Algebra, World Scientific Publishing.

R.Majoob, On Weakly Prime L-Ideals, Italian Journal of Pure and Applied Mathematics-N.

36-2016(465-472).

6. Zadeh L. A., Fuzzy sets, Inform. and Control, 8, 338–353, 1965.

7. Zahedi M. M., A Note On L-Fuzzy Primary and Semiprime Ideals, Fuzzy Sets and Systems

51 (2): 243-247, 1992.

52



1School of Mathematics‎, ‎Statistic and Computer Sciences‎, ‎University of Tehran 2Department of Mathematics‎, ‎Payame Noor University‎, ‎P.O.Box 19395-3697‎, ‎Tehran‎, ‎Iran


Some Results on Tensor Product of Krasner Hypervector Spaces

R‎. ‎Ameri1, ‎K‎. ‎Ghadimi

2

We introduce and study tensor product of Krasner hypervector spaces (Krasner hyperspac

es)‎. ‎Here we introduce the (resp‎. ‎multivalued) middle linear maps of Krasner hyperspaces and

construct the categories of linear maps and multivalued linear maps of Krasner hyperspaces‎. ‎It

is shown the tensor product of two Krasner hypespaces‎, ‎as an initial object in this

category‎, ‎exists‎. ‎Also‎, ‎notion of a quasi-free object in category of Krasner hyperspaces are

introduced and it is proved that in this category a quasi-free object up to maximum is unique‎.

Keyword(s): Krasner hypervector space‎, ‎Multivalued middle linear map‎, Quasi-free‎, Tensor

product

2010 AMS Classification: ‎20N20

1 Introduction

The theory of algebraic hyperstructures is a well-established branch of classical algebraic

theory. Hyperstructure theory was first proposed in 1934 by Marty, who defined hypergroups

and began to investigate their properties with applications to groups, rational fractions and

algebraic functions [10]. It was later observed that the theory of hyperstructures has many

applications in both pure and applied sciences; for example, semihypergroups are the simplest

algebraic hyperstructures that possess the properties of closure and associativity. The theory

of hyperstructures has been widely reviewed ([6], [7], [8], [9] and [12]) (for more see [1, 2, 3,

4, 5]).

In [11] M. Motameni et. al. studied hypermatrix. R. Ameri in [1] introduced and studied

categories of hypermodules. Let 𝑉 and 𝑊 be two Krasner hyperspaces over the hyperfiled 𝐾.

The purpose of this paper is the study of tensor product of Krasner hyperspaces. We introduce

the category of multivalued linear maps of Krasner hyperspaces and then construct the tensor

product of 𝑉 and 𝑊 as initial object in this category.

2 Preliminaries and main results

Let 𝐻 be a nonempty set. A map ∙ ∶ 𝐻 × 𝐻 → 𝑃∗(𝐻) is called hyperoperation or join

operation, where 𝑃∗(𝐻) is the set of all nonempty subsets of 𝐻 . The join operation is

extended to nonempty subsets of 𝐻 in natural way, so that 𝐴 ∙ 𝐵 is given by

53





𝐴 ∙ 𝐵 = ⋃{𝑎 ∙ 𝑏 | 𝑎 ∈ 𝐴 𝑎𝑛𝑑 𝑏 ∈ 𝐵}.

the notations 𝑎 ∙ 𝐴 and 𝐴 ∙ 𝑎 are used for {𝑎} ∙ 𝐴 and 𝐴 ∙ {𝑎} respectively. Generally, the

singleton {𝑎} is identified by its element 𝑎.

Definition 1. [7] A semihypergroup (𝐻, +) is called a canonical hypergroup if the following

conditions are satisfied:

(i) 𝑥 + 𝑦 = 𝑦 + 𝑥 for all 𝑥, 𝑦 ∈ 𝑅;

(ii) There exists 0 ∈ 𝑅 (unique) such that for every 𝑥 ∈ 𝑅, 𝑥 ∈ 0 + 𝑥 = 𝑥;

(iii) For every 𝑥 ∈ 𝑅, there exists a unique element, say 𝑥′ such that 0 ∈ 𝑥 + 𝑥′

(we denote 𝑥′ = −𝑥);

(iv) For every 𝑥, 𝑦, 𝑧 ∈ 𝑅, 𝑧 ∈ 𝑥 + 𝑦 ⟺ 𝑥 ∈ 𝑧 − 𝑦 ⟺ 𝑦 ∈ 𝑧 − 𝑥;

from the definition it can be easily verified that −(−𝑥) = 𝑥 and −(𝑥 + 𝑦) = −𝑥 − 𝑦.

Definition 2. [7] Let (𝐾, +, ∗) be a hyperfield and (𝑉, +) be a canonical hypergroup. We

define a Krasner hyperspace over K to be the quadruplet (V, +, ∙, K) where ∙ is a single-

valued operation

∙ ∶ K × V → V

such that for all a ∈ K and x ∈ V we have a ∙ x ∈ V , and for all a, b ∈ K and x, y ∈ V the

following conditions are satisfied:

(H1) a ∙ (x + y) = a ∙ x + a ∙ y;

(H2) (a + b) ∙ x = a ∙ x + b ∙ x;

(H3) a ∙ (b ∙ x) = (a ∗ b) ∙ x;

(H4) 0 ∙ x = 0;

(H5) 1 ∙ x = x.

Remark 1. For simplicify, we say 𝑉 is a 𝐾𝑟-hyperspace.

54





Definition 4. Let (𝐹, ·) be an object in the category 𝐾𝑟 − H𝑣𝑒𝑐𝑡 and 𝑖 ∶ 𝑋 ↪ 𝐹 be an

inclusion map of sets. We say that 𝐹 is quasi-free on the subset X provided that:

(i) 𝐹 = ⟨𝑋⟩;

(ii) For any object 𝑉 in 𝐾𝑟 − 𝐻𝑣𝑒𝑐𝑡 and any multivalued map 𝜆 ∶ 𝑋 → 𝑃∗(𝑉 ), there is a

maximum 𝑠𝑚𝑣, �̅� ∶ 𝐹 → 𝑃∗(𝑉 ) such that for all 𝑥 ∈ 𝑋, we have �̅�𝑖(𝑥) = 𝜆(𝑥).

Definition 5. Let 𝑉 and 𝑊 be two 𝐾𝑟-hyperspaces over a hyperfield 𝐾 . Let F be the free

abelian group on the set 𝑉 × 𝑊. Let 𝐻 be the subgroup of 𝐹 generated by all elements of the

following forms (for all 𝑣, 𝑣′ ∈ 𝑉, 𝑤, 𝑤′ ∈ 𝑊, and 𝑎 ∈ 𝐾):

(i) (𝑣 + 𝑣′, 𝑤) − (𝑣, 𝑤) − (𝑣′, 𝑤), where (𝑣 + 𝑣′, 𝑤) =∪𝑡∈𝑣+𝑣′ (𝑡, 𝑤)

(ii) (𝑣, 𝑤 + 𝑤′) − (𝑣, 𝑤) − (𝑣, 𝑤′);

(iii) (𝑎 · 𝑣, 𝑤) − (𝑣, 𝑎 · 𝑤).

The quotient group 𝐹/𝐻 is called a tensor product of 𝑉 and 𝑊; it is denoted 𝑉 ⊗𝐾 𝑊. The

coset (𝑣, 𝑤) + 𝐾 of the element (𝑣, 𝑤) in 𝐹 is denoted 𝑣 ⊗ 𝑤 ; the coset of (0, 0) is

denoted 0.

Theorem 1. Let 𝐹 be a 𝐾𝑟-hyperspace over a hyperfield 𝐾 and 𝑋 be a basis for 𝐹. Then

(i) If 𝑗 ∶ 𝑋 ↪ 𝐹 is a inclusion map, then for all 𝐾𝑟-hyperspace 𝑉 and map f ∶ 𝑋 → 𝑃∗(𝑉 ),

there is a maximum 𝑠𝑚𝑣, 𝜑 ∶ 𝐹 → 𝑃∗(𝑉 ) such that the diagram

is commutative.

(ii) For all 𝐾𝑟-hyperspace 𝑉 and f ∶ 𝑋 → 𝑃∗(𝑉 ) induced maximum 𝑠𝑚𝑣, 𝜑 ∶ 𝐹 → 𝑃∗(𝑉 ),

means there is a maximum 𝑠𝑚𝑣, 𝜑 ∶ 𝐹 → 𝑃∗(𝑉 ) such that 𝜑|𝑋 = 𝑓.

55





Given hyperspaces 𝑉 and 𝑊 over a hyperfield 𝐾, it is easy to verify that the map 𝑖 ∶ 𝑉 ×

𝑊 → 𝑉 ⊗𝐾 𝑊 given by (𝑣, 𝑤) ⟼ 𝑣 ⊗ 𝑤 is a middle linear map. The map i is called the

canonical middle linear map. Its importance is seen in

Theorem 2. Let 𝑉 and 𝑊 be 𝐾𝑟-hyperspaces over a hyperfield 𝐾, and let 𝑍 be an abelian

group. If 𝑔 ∶ 𝑉 × 𝑊 → 𝑍 is a middle linear map, then there exists unique group

homomorphism �̅� ∶ 𝑉 ⊗𝐾 𝑊 → 𝑍 such that �̅�𝑖 = 𝑔 , where 𝑖 ∶ 𝑉 × 𝑊 → 𝑉 ⊗𝐾 𝑊 is the

canonical middle linear map. 𝑉 ⊗𝐾 𝑊 is uniquely determined up to isomorphism by this

property. In other words 𝑖 ∶ 𝑉 × 𝑊 → 𝑉 ⊗𝐾 𝑊 is universal in the category ℳℒ(𝑉, 𝑊) of all

middle linear maps on 𝑉 × 𝑊.

Reference(s):

1. Ameri R‎.‎, ‎On categories of hypergroups and hypermodules‎, ‎Journal of Discrete

Mathematical Sciences and Cryptography‎, ‎‎6‎, ‎2-3, 121-132, 2003.

2. ‎Ameri R‎.‎, ‎Dehghan O.R.‎, ‎On dimension of hypervector spaces‎, ‎European Journal of Pure

and Applied Mathematics‎, ‎‎1‎, ‎‎2, 32-50,‎2008.

3. Ameri R‎.‎, ‎Borzooei ‎R.A‎., ‎Ghadimi‎ K‎., ‎Representations of polygroups‎, ‎Italian Journal of

Pure and Applied Mathematics, 37, 595-610, 2016‎.

4. Ameri R‎.‎, ‎ Borzooei ‎R.A‎., ‎Ghadimi‎ K‎., ‎Multivalued linear transformations of

hyperspaces‎, ‎Ratio Mathematica‎, ‎27, 37-47,‎2014.

5. Ameri R‎.‎, Ghadimi‎ K‎., Borzooei ‎R.A‎., ‎ ‎Multivalued linear transformations in categories of

hypervector spaces,‎‎(submitted)‎.

6. ‎Corsini P‎.‎, ‎Prolegomena of Hypergroup Theory‎, ‎Second Edition‎, ‎Aviani Editor‎, ‎1993‎.

7. ‎Corsini P‎.‎, ‎‎Leoreanu-Fotea V‎.‎, ‎Applications of Hyperstructure Theory‎, ‎Kluwer Academic

Publishers‎, ‎Dordrecht‎, ‎Hardbound‎, ‎2003‎.

8. ‎Davvaz B‎.‎, ‎Polygroup Theory and Related Systems‎, ‎World Scientific‎, ‎2013‎.

9. ‎Davvaz B‎.‎, ‎‎Leoreanu-Fotea‎ V‎., ‎Hyperring Theory and Applications‎, ‎International

Academic Press‎, ‎USA‎, ‎2007‎.

‎10. ‎Marty‎ F‎., Sur une g ́en ́eralisation de la notion de groupe. In 8`eme congr`es des

Math ́ematiciens Scandinaves, Stockholm ‎, 45-49, 1934.

11. ‎Motameni‎ M‎., ‎Ameri R‎., ‎Sadeghi R‎.‎, Hypermatrix based on Krasner hypervector spaces‎,

Ratio Mathematica, 25, 77-94,‎2013.

12. ‎Vougiouklis T‎.‎, ‎Hyperstructures and Their Representations‎, ‎Hadronic Press‎, ‎Inc.‎, 115,

Palm Harber, USA, ‎1994‎.

56



1Department of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili

E-mail(s): Jafar A’zami, [email protected], [email protected]

Fuzzy coprimary submodules and their representation

Jafar A'zami1

Let R be a commutative ring with non-zero identity and let M be a non-zero unitary R-

module. The concept of fuzzy coprimary submodule as a dual notion of fuzzy primary will be

studied. Among other things, the behavior of this concept with respect to fuzzy localization

formation and fuzzy quotient will be examined. Also the uniqueness theorem for a non-zero

fuzzy representable submodule of M will be proved.

Keywords: Fuzzy coprimary submodule, Fuzzy prime and primary ideal, Fuzzy

localization, fuzzy coprimary representation, fuzzy attached primes.

2010 AMS Classification: 08A72

Reference(s):

1. Kirby D. Coprimary decomposition of Artinian modules,J. London Math. Soc, 6, 571–

576, 1973.

2. Macdonald I. G., Secondary representation of modules over a commutative ring, Symposia.

Math. 11, 23–43, 1973.

3. Mordeson J. N., and Malik D. S., Fuzzy Commutative Algebra, World Scientific, 1998.

57



1AmasyaUniversity, Department of Mathematics, Amasya, Turkey


Constructing Topological Hyperspace with Soft Sets

Güzide Şenel1

In this study, by defining soft ditopological spaces, I construct a topological

hyperspace with soft sets. I make a new approach to the soft topology via soft set theory, with

defining two structures on a soft set - a soft topology and a soft subspace topology. Moreover,

I characterize separation axioms in soft ditopological spaces and investigate the relations

between soft topological and soft ditopological structures [10]. Based on this idea, the

relations between the separation axioms of ordered soft topological spaces and the separation

axioms of the corresponding soft ditopological spaces are established.

Keywords: Soft sets, hyperspace, topological hyperspace

2010 AMS Classification: 54B20

References:

1. Ali, M.I., Feng, F., Liu, X., Min, W.K., Shabir, M., On some new operations in soft set theory,

Computers and Mathematics with Applications 57, 1547-1553, 2009.

2. Aygünoglu, A. Aygün, H., Some notes on soft topological spaces, Neural Computation and

Application, 2011.

3. Aktaş, H. ve Cagman, N., Soft sets and soft groups, Information Sciences, 177(1), 2726-2735, 2007.

4. Cagman, N., Enginoglu, S., Soft set theory and uni-int decision making, European Journal of

Operational Research 207, 848-855, 2010.

5. Dizman, T. et al., Soft Ditopological Spaces, Filomat 30:1, 209–222, 2016.

6. Ittanagi, B.M., Soft Bitopological Spaces, International Journal of Computer Applications

(0975 8887), Volume 107 - No. 7, December 2014.

7. Maji, P.K., Biswas, R., Roy, A.R., An Application of Soft Sets in A Decision Making Problem,

Computers and Mathematics with Applications 44, 1077- 1083, 2002.

8. Maji, P.K., Biswas, R., Roy, A.R., Soft set theory, Computers and Mathematics with Applications

45, 555-562, 2003.

9. Molodtsov, D.A., Soft set theory-first results, Computers and Mathematics

with Applications 37, 19-31, 1999.

10. Şenel, G., The Theory of Soft Ditopological Spaces, International Journal of Computer

Applications (0975 - 8887), Volume 150 - No.4, September 2016.

11. Şenel, G., Çağman, N., Soft Topological Subspaces, Annals of Fuzzy Mathematics and

Informatics, vol.10, no : 4, 525 - 535, 2015.

12. Shabir, M., Naz, M., On soft topological spaces, Computers and Mathematics with Applications

61, 1786-1799, 2011.

58



1 Yildiz Technical University, Department of Mathematics, Istanbul, Turkey

E-mails: [email protected], [email protected], [email protected] , [email protected]

Fuzzy Weakly Prime Γ-Ideal in Γ-Rings

Gülşah Yeşilkurt1, Serkan Onar

1, Deniz Sönmez

1 and Bayram Ali Ersoy

1

In this study, we investigate fuzzy weakly prime, fuzzy partial weakly prime and

fuzzy weakly semiprime Γ- ideal of a Γ-ring. We obtain some characterizations of fuzzy

weakly prime, fuzzy partial weakly and fuzzy weakly semiprime Γ-ideal of a Γ-ring. First we

give the definition of fuzzy weakly prime ideal, fuzzy weakly semiprime and fuzzy partial

weakly prime Γ- ideal. Further we give some properties of its.

Keywords: Fuzzy prime ideal, fuzzy weakly prime Γ-ideal, fuzzy partial weakly prime Γ-ideal,

fuzzy weakly semiprime Γ-ideal.

2010 AMS Classification: 03E72, 16D25, 16P99, 08A72.

References:

1. Mahjoob R., On Weakly Prime L-Ideals, Italian Journal of Pure and Applied Math., 36, 465-472,

2016

2. Zahedi, MM., A Characterization of L-Fuzzy Prime Ideals, Fuzzy Sets and Systems, 44 (1): 147-

160, 1991,

3. Rao, M. M. K., Fuzzy Prime Ideals in Ordered Γ-Semiring, Journal of International Mathemati-

cal Virtual Instutute, 7, 85–99, 2017

4. Mordeson, J.N. and Malik, D. S., Fuzzy Commutative Algebra, World Scientific Publishing Co.

Pte. Ltd., 1998

5. T. K. Dutta and T. Chanda, Fuzzy Prime Ideals in -rings, Bull. Malays. Math. Sci. Soc. (2) 30(1)

(2007), 65–73.

6. W. E. Barnes, On the -rings of Nobusawa, Pacific J. Math. 18(1966), 411–422.

7. T. K. Dutta and T. Chanda, Structures of fuzzy ideals of �-ring, Bull. Malays. Math. Sci.Soc. (2)

28(1)(2005), 9–18.

8. S. Kyuno, Prime ideals in gamma rings, Pacific J. Math. 98(2)(1982), 375–379.

9. Zadeh, L.A., Fuzzy Sets, Inform. and Control, 8 (1965), 338-353.

10.Anderson, D.D., Smith, E., Weakly prime ideals, Houston J. of Math., 29 (4)(2003), 831-840.

11. Ebrahimi Atani, Sh., Farzalipour, F., On weakly primary ideals, Georgian Mathematics Journal, 12

(3) (2005), 423-429.

59






1YildizTechnicalUniversity,Department of Mathematics, Istanbul, Turkey

E-mail(s):[email protected] , [email protected] , [email protected] , [email protected]

Intuitionistic Fuzzy 2-Absorbing Ideals of Commutative Rings

Sanem YAVUZ1, Serkan ONAR

1, Deniz SONMEZ

1, Bayram Ali ERSOY

1

The aim of this paper is to give a definitions of intuitionistic fuzzy 2- absorbing ideals

and intuitionistic fuzzy weakly completely 2- absorbing ideals of commutative rings and to

give their properties. Moreover, we give diagram which transition between definitions of

intuitionistic fuzzy 2- absorbing ideals of commutative rings.

Keywords: Fuzzy Set, Intuitionistic Fuzzy Set, Intuitionistic Fuzzy Ideal, Intuitionistic

Fuzzy Prime Ideal, 2-Absorbing Ideal, Intuitionistic Fuzzy Completely Prime Ideal,

Intuitionistic Fuzzy Weakly Completely Prime Ideal, Intuitionistic Fuzzy K- Prime Ideal,

Intuitionistic Fuzzy 2-Absorbing Ideal, Intuitionistic Fuzzy Strongly 2-Absorbing Ideal,

Intuitionistic Fuzzy Weakly Completely 2- Absorbing Ideal, Intuitionistic Fuzzy K-2-

Absorbing Ideal.


References:


2. Atanassov K., Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20, 87-96, 1986.

3. Hur K., Kang H.W. and.Song H.K, Intuitionistic fuzzy subgroups and subrings, Honam

Math J., 25, 19-41 , 2003.

4. Marashdeh M.F., Salleh A.R., Intuitionistic fuzzy rings, International Journal of Algebra, 5,

37-47 , 2011.

5. Badavi A., On 2- absorbing ideals of commutative rings, Bull. Austral. Math. Soc., 75,

417-429 , 2007.

6. Darani A.Y., On L- fuzzy 2- absorbing ideals, Italian J. of Pure and Appl. Math., 36, 147-

154 , 2016.

7. Bakhadach I., Melliani S., Oukessou M. and Chadli L.S., Intuitionistic fuzzy ideal and

intuitionistic fuzzy prime ideal in aring, Intuitionistic Fuzzy Sets (ICIFSTA), 22, 59-63, 2016.

8. Malik D.S. and Mordeson J.N., Fuzzy Commutative Algebra, World Scientific Publishing,

1998.

60






1Yildiz Technical University, Department of Mathematic Education, Istanbul, Turkey 2Yildiz Technical University, Department of Mathematics, Istanbul, Turkey

E-mail: [email protected] , [email protected]

Transition from Two-Person Zero-Sum Games to Cooperative Games with Fuzzy

Payoffs

Adem C. Cevikel 1 and Mehmet Ahlatcioglu

2

In this study, we deal with games with fuzzy payoffs. We proved that players who are

playing a zero-sum game with fuzzy payoffs against nature are able to increase their joint

payoff, and hence their individual payoffs by cooperating. It is shown that, a cooperative

game with the fuzzy characteristic function can be constructed via the optimal game values of

the zero-sum games with fuzzy payoffs against nature at which players' combine their

strategies and act like a single player. It is also proven that, the fuzzy characteristic function

that is constructed in this way satisfies the superadditivity condition. Thus we considered a

transition from two-person zero-sum games with fuzzy payoffs to cooperative games with

fuzzy payoffs. The fair allocation of the maximum payoff (game value) of this cooperative

game among players is done using the Shapley vector.

Keywords: Cooperative game; Fuzzy number; Fuzzy games; Shapley vector.

2010 AMS Classification: 91A10, 91A12, 03E72.

References:

1. L.A.Zadeh, Fuzzy sets, Information and control, 8, (1965), 338-353.

2.H.J.Zimmermann, Fuzyy sets, Decision making, and expert systems, Kluwer academic publishers,

Boston, (1991).

3. D.Butnariu, Fuzzy games: A description of the concept, Fuzzy Sets and Systems 1 (1978) 181-192.

4. L.Campos, Fuzzy linear programming models to solve fuzzy matrix games, Fuzzy Sets and Systems

32 (1989) 275-289.

5. Nishizaki and M.Sakawa, Equilibrium Solutions in Multiobjective Bimatrix Games with Fuzzy

Payoffs and Fuzzy Goals, Fuzzy Sets and Systems, 111 (2000) 99-116.

6. M.Sakawa and I. Nishizaki, A solution concept in multiobjective matrix game with fuzzy payoffs

and fuzzy goals, Fuzzy logic and its applications to engineering, Information science, (1995), 417-426.

7. A.C.Cevikel and M.Ahlatçıoğlu, Solutions for fuzzy matrix games, Computers & Mathematics with

Applications, Vol.60,3, (2010), 399-410.

8. D.F.Li, A fuzzy multiobjective approach to solve fuzzy matrix games, The journal of fuzzy

mathematics, 7, (1999), 907-912.

9. D.F.Li, Fuzzy constrained matrix game with fuzzy payoffs, The journal of fuzzy mathematics, 7,

(1999), 873-880.

10. M.Sakawa and I. Nishizaki, Max-min solutions for fuzzy multiobjective matrix games, Fuzzy Sets

and Systems, 67 (1994) 53-69.

61




1Department of Mathematics‎, ‎University of Mazandaran‎, ‎Babolsar‎, ‎Iran 2School of Mathematics‎, ‎Statistic and Computer Sciences‎, ‎University of Tehran‎, ‎Tehran‎, ‎iran‎ 3Department of Mathematics‎, ‎University of Mazandaran‎, ‎Babolsar‎, ‎Iran


On computation of fundamental group of a finite hypergroup

K‎. ‎Abbasi1, R‎. ‎Ameri

2, Y‎. ‎Talebi-Rostami

3

‎ The purpose of this paper is the computation of fundamental group in a finite

hypergroup ).,( H In this regards we first obtain an algorithm to construct the equivalence

classes of ,* ‎ the fundamental relation on ,H ‎ then we construct ),,/( * H ‎the fundamental

group of .H In particular‎, ‎given some classes of hypergroups‎, ‎we find fundamental groups of

them‎. ‎We apply a comprehensive Java program to compute the fundamental group of a given

finite hypergroup. ‎‎It consists of two sub-programs (Hypergroup generator and Main)‎.

hypergroup generator‎, ‎counts all hypergroups of order )(3 Nnn and isomorphism classes

of them‎ and ‎ enumerates quasihypergroups of order ‎‎n‎‎ and all -equivalence classes and by

the next sub-program (Main)‎, ‎ it is checked that is an arbitrary hypergroupoid ),( H of order

)( Nnn is a hypergroup or not‎. ‎If it is a hypergroup‎, ‎this sub-program computes its -

equivalence classes and fundamental group‎.

Keywords: ‎Hypergroup‎, ‎Fundamental relation‎, ‎Fundamental group, Computation.

2010 AMS Classification: ‎20N20‎, ‎68W30‎, ‎68W40.

Acknowledgements:

The author partially has been supported by "Algebraic Hyperstructure Excellence (AHETM),

Tarbiat Modares University, Tehran, Iran" and "Research Center in Algebraic hyperstructures

and Fuzzy Mathematics, University of Mazandaran, Babolsar, Iran".

References:

[1] Ameri R.‎, ‎Nouzari T‎., ‎A new characterization of fundamental relation on hyperrings‎,

Int‎. ‎J‎. ‎‎Contemp‎. ‎Sciences, 5(10), 721-738, 2010.

[2] Bayon R.‎, ‎ ‎Lygeros N.‎, ‎‎Advanced results in enumeration of hyperstructures‎, Journal of

Algebra, 320, 821-835‎, 2008.

[3] Corsini P.‎, ‎Prolegomena of Hypergroup Theory‎, second edition‎., ‎Aviani Editor, 1993.

[4]Corsini P., Leoreanu‎ ‎V‎‎.‎, ‎Applications of Hyperstructures Theory‎, ‎Advanced in

Mathematics‎, ‎Kluwer Academic Publishers, 2003.

62



1Department of Mathematics‎, ‎University of Mazandaran‎, ‎Babolsar‎, ‎Iran 2School of Mathematics‎, ‎Statistic and Computer Sciences‎, ‎University of Tehran‎, ‎Tehran‎, ‎iran‎ 3Department of Mathematics‎, ‎University of Mazandaran‎, ‎Babolsar‎, ‎Iran


[5] Freni D.‎‎, ‎A new characterization of the derived hypergroup via strongly regular

equivalences,,‎Communication in algebra, 30(8), 3977-3989, 2002.

[6] Freni D‎., ‎On a Strongly Regular Relation in Hypergroupoids, Pure Math‎. ‎Appl‎, ‎Ser‎. ‎A, 3-4

191-198, 1992.

[7] Freni D.‎‎, ‎Une note sur le coeur d'un hypergroupe et sur la cloture transitive ,* de , A

note on the core of a hypergroup and the transitive closure * of , ‎Rivista‎. ‎di Mat‎. ‎Pura Appl

8, 153-156‎, 1991.

[8] Koskas M‎.‎, ‎Groupoides‎, ‎Demi-hypergroupes et hypergroupes‎, J‎. ‎Math‎. ‎Pures Appl, 49,

155-192‎, 1970.

[9] Marty F‎.‎, ‎Sur une generalization de la notion de groupe‎, in‎ th8 ‎Congress

Math‎. ‎Scandinaves‎, ‎Stockholm‎, ‎Sweden, 45-49‎, 1934.

[10] Vougiouklis T‎.‎, ‎The fundamental relations in hyperrings‎. ‎The general hyperfield‎,

Proceeding of th4 International congress in Algebraic Hyperstructures and Its Applications

AHA, 1990.

63



1School of Mathematics, Statistics and Computer Science, University of Tehran, Tehran, Iran


Various Kinds of Quotient of a Canonical Hypergroup

Hossein Shojaei1 and Reza Ameri

1

In this study various kinds of quotients of a given canonical hypergroup are introduced

and studied. In this regards we introduce some regular equivalence relations on a canonical

hypergroup to construct different quotients for this hyperstructure. We will proceed to

investigate the relationships among these relations such that these extracted quotient

structures be equal. Also, the relationship between the heart of a give canonical hypergroup

and its quotient via an equivalence relation is studied and some related basic results are

obtained. Finally, we study the quotient hyperstructures of a canonical hypergroup induced

via a normal canonical subhypergroup, and show that for this special kind of quotient space

all various kinds of quotients are concid.

Keywords: Hypergroup, Canonical Hypergroup, Quotient Hypergroup, Heart.

2010 AMS Classification: 20N20, 20N15

References:

1. R. Ameri, On categories of hypergroups and hypermodules, Journal of Discrete

Mathematical Sciences and Cryptography, 6(2-3) (2003), 121--132.

2. P. Corsini, Prolegomena of Hypergroup Theory, Second edition, Aviani editore, Tricesimo,

1993.

3. P. Corsini and V. Leoreanu, Applications of Hyperstructures Theory, Advanced in

Mathematics, Kluwer Academic Publisher, Dordrecht, 2003.

4. B. Davvaz, Polygroup Theory and Related Systems, World Scienti c, 2013.

5. B. Davvaz, V. Leoreanu-Fotea, Hyperring Theory and Applications, International

Academic Press, USA, 2007.

6. D. Freni,Une note sur le cur d'un hypergroupe et sur la cloture transitive, Riv. Mat. Pura

Appl. 8 (1991), 153-156.

7. F. Marty, Sur une generalization de la notion de groupe, 8th Congress Math. Scandenaves,

Stockholm, (1934), 45-49.

64



1School of Mathematics, Statistics and Computer Science, University of Tehran, Tehran, Iran


8. M. Mittas, Hypergroupes canoniques, Math. Balkanica, 2, (1972), 165-179.

9. H. Shojaei, R. Ameri, On hypergroups with trivial fundamental group, in: 46th Annual

Iranian Mathematics Conference, Yazd University, (2015), 238-241.

10. M. Velrajan, A. Arjunan, Note on isomorphism theorems of hyperrings, Int. J. Math. and

Math. Sci (2010), Article ID 376985.

11. T. Vougiouklis, Hyperstructures and their Representations, Hadronic

Press, Inc., Palm Harber, USA, 1994.

65



1Ege University, Institute of Science, Department of Mathematics, Izmir, Turkey )

2Ege University, Department of Mathematics, Izmir, Turkey)


On multipliers of hyper BCC-algebras

Didem SÜRGEVİL UZAY1, Alev FIRAT

2

In this paper, we introduced the notion of multiplier of a hyper BCC-algebra, and

investigated some properties of hyper BCC- algebras. And then we introduced notion of

kernels. Also we gave some propositions related with isotone and Fixd(H).

Keywords: hyper BCC-algebra, multiplier, isotone, Fixd(H), regular.

2010 AMS Classification: 20N20, 16W25

Reference(s):

1. Marty F., Surene generalization de la notion de group, In eigth Congress Math.,

Scandinaves, Stockholm, 45-49, 1934.

2. Iseki K., An algebra related with propositional calculus, On hyper BCK algebras,

Italian Journal of Pure and Applied Msthematics, 8, 127-136, 2000.

3. Komori Y., The class of BCC-algebras is not a variety, Mathematica Japonica

29(3), 391-394, 1984.

4. Jun Y.B., Zahedi M.M., Xin X. L. Borzooei R.A., On hyper BCK-algebras,

Italian Journal of Pure and Applied Mathematics, 8, 127-136, 2000.

5. . Jun Y.B., Xin X. L., Zahedi M.M and Roh E.H., Strong hyper BCK-ideals of hyper

BCK-algebras,Math. Japonica 51(3), 493-498, 2000.

6. . Borzooei R.A., Dudek, W.A., Koohestani N., On hyper BCC-algebras, Hindawi

Publishing Corporation, International Journal of Math. Sci., 1-18, 2006.

7. Kim K.H., Lim H.J., On Multipliers of BCC-algebras, Honam Mathematical Jour-

nal J. 35(2), 201-210, 2013.

66





1Yaşar University, Department of Mathematics, Izmir, Turkey

2Ege University, Faculty of Science, Department of Mathematics,Izmir, Turkey


Derivations on Hyperlattices

Şule Ayar ÖZBAL1, Alev FIRAT

2

In this paper, we introduced the notion of derivations on hyperlattices and investigated

some related properties. Also, we characterized the Fixd(L), and Kerd(L) by derivations

Keyword(s): Lattices, hyperlattices, derivations.

2010 AMS Classification: 03G10, 20N20, 16W25

References:

1. Yon Y.H. , Kim K.H., On Expansive Linear Maps and V-multipliers of Lattices ,

Quaestiones Mathematicae, 33:4, 417-427.

2. Szasz G., Derivations of Lattices, Acta Sci. Math. (Szeged) 37 (1975), 149-154.

3. Szasz G., Translationen der Verbande, Acta Fac. Rer. Nat. Univ. Comenianae 5 (1961),

53-57.

4. Szaz A., Partial Multipliers on Partiall Ordered Sets, Novi Sad J. Math. 32(1) (2002), 25-

45.

5. Szaz A. And Turi J. , Characterizations of Injective Multipliers on Partially Ordered Sets,

Studia Univ. "BABE-BOLYAI" Mathematica XLVII(1) (2002), 105-118.

6. Posner E.C., Derivations in prime rings, Proc. Amer. Math. Soc. 8,(1957), 1093-1100.

7. Marty F., Surene generalization de la notion de group, In eigth Congress Math.,

Scandinaves, Stockholm, (1934), 45-49.

8. James J., Transposition: Noncommutative Join Spaces, Journal of Algebra, 187(1997),97-

119.

9. Rosaria R., Hypera ne planes over hyperrings, Discrete Mathematics, 155,(1996),215-223.

10. Xin X. L., BCI-algebras, Disccussion Mathematicae General Algebra and Applications,

26(2006),5-19.

11. Rasouli S. and Davvaz B., Lattices Derived from Hyperlattices, Communivations in

Algebra, 38:8, (2010), 2720-2737.

12. Ozbal S. A, Firat A. (2010). Symmetric Bi-Derivations of Lattices. Ars Combinatoria,

97(4), 471-477.

13. Ferrari L., On Derivations of Lattices, Pure Math. Appl. 12 (2001), no.4, 365-382.

14. Xin X. L., Li T.Y. and Lu J. H., On derivations of Lattices, Information Sciences 178,

(2008) 307-316.

67



1University of Mohaghegh Ardabili, Ardabil, Iran


On fuzzy -prime ideals

Naser Zamani1

Let R be a commutative ring with identity. Let FI(R) be the set of all fuzzy ideals of R

and : 0RFI R FI R be a function. We introduce the concept of fuzzy -prime

ideals. Some relationships between fuzzy -prime ideals and prime ideals of R will be

investigated. We find conditions under which fuzzy -primness gives primness and vice

versa. Also, the behaviour of this concept in rings product will be studied.

Keywords: Fuzzy -prime ideals, fuzzy prime ideal


References:

68



1Department of Mathematics, Faculty of Science, , Khon Kaen University, Khon Kaen 40002, Thailand

2Department of Mathematics, Yazd University,Yazd, Iran


On pure hyperideals in ordered semihypergroups

Thawhat Changphas 1 and Bijan Davvaz 2

In this study, the notions of pure hyperideal, weakly pure hyperideal and purely prime

hyperideal in ordered semihypergroups are introduced and studied. We prove that the set of

all purely prime hyperideals is topologized.

Keywords: Algebraic hyperstructure, ordered semihypergroup, weakly regular, püre

hyperideal, weakly pure hyperideal, purely prime hyperideal, topology


References:

69



1 Department of Mathematics, Maku Branch, Islamic Azad University, Maku, Iran

2 Department of Mathematics, Payame Noor University, p. o. box. 19395-3697, Tehran, Iran. 3 Department of Mathematics, Shahid Beheshti University, Tehran, Iran

E-mail(s): [email protected] , [email protected], [email protected]

Relation Between Hyper K-algebras and Superlattice

(Hypersemilattice)

A.Rezazadeh1, A. Radfar

2, R. A. Borzooei

3

In this paper, by considering the notions of hypersemilattice and superlattice, we prove

that any commutative and positive imolicative hyper K-algebra, is a hypersemilattice.

Moreover, we prove that any bounded commutative hyper K-algebra with condition L is a

superlattice.

1. Introduction

The theory of hyperstructures was introduced in 1934 by Marty [5]. This theory has been

subsequently developed by the contribution of various authors. In [1], R. A. Borzooei et al.

applied the hyperstructures to K-algebras and introduced the notion of a hyper K-algebra and

investigated some related properties. Some researchers applied the hyperstructure to some

accepts of lattice theory and the notion of hypersemilattice was introduced by Z. Bin et al. in

[2] and the notion of superlattice was introduced by Mittas and Konstantinidou in [6]. In this

paper, we prove that every hyper K-algebra by some condition is a hypersemilattice. In

follow, we introduce the notions ∧ and ∨ on hyper K-algebras and we prove that every hyper

K-algebra

of order 3 by some condition is a superlattice.

2. Preliminary

In this section, we give some definitions and theorems that we need in the next sections.

Definition 2.1. [2] Let L be a nonempty set with a binary hyperoperation ⊗ on L such that for

all a, b, c ∈ L,

the following condition hold:

(i) a ∈ a ⊗ a,

(ii) a ⊗ b = b ⊗ a,

(iii) (a ⊗ b) ⊗ c = a ⊗ (b ⊗ c).

Then (L, ⊗) is called a hypersemilattice.

Definition 2.2. [6] A superlattice is a partially ordered set (S, <) with two hyper operations ∨

and ∧ such that the following properties hold:

(S1) a ∈ a ∨ a and a ∈ a ∧ a,

(S2) a ∨ b = b ∨ a and a ∧ b = b ∧ a

70









(S3) (a ∨ b) ∨ c = a ∨ (b ∨ c) and (a ∧ b) ∧ c = a ∧ (b ∧ c),

(S4) a ∈ a ∨ (a ∧ b) and a ∈ a ∧ (a ∨ b),

(S5) if a < b then (b ∈ a ∨ b and a ∈ a ∧ b),

(S6) (b ∈ a ∨ b or a ∈ a ∧ b) implies a < b.

for all a, b, c ∈ S

Definition 2.3. [1] By a hyper K-algebra we mean a nonempty set H endowed with a

hyperoperation ”◦” and a constant 0 satisfy the following axioms:

(HK1) (x ◦ z) ◦ (y ◦ z) < x ◦ y,

(HK2) (x ◦ y) ◦ z = (x ◦ z) ◦ y,

(HK3) x < x,

(HK4) x < y and y < x imply x = y,

(HK5) 0 < x.

for all x, y, z ∈ H, where x < y is defined by 0 ∈ x ◦ y and for every A, B ⊆ H, A < B is defined

by ∃a ∈ A, ∃b ∈ B such that a< b.

Theorem 2.4. [1] Let H be a hyper K-algebra. Then the following are hold:

(i) x ∈ x ◦ 0,

(ii) x ◦ y < z ⇔ x ◦ z < y,

(iii) x ◦ (x ◦ y) < y,

(iv) x ◦ y < x,

(v) A ◦ B < A.

for all x, y, z ∈ H.

3. Relation between hyper K-algebras and hypersemilattice

In this section we prove that every commutative and positive implicative hyper K-algebra is a

hypersemilattice.

Definition 3.1. A hyper K-algebra (H, ◦,0) is called commutative if for all x, y ∈ H,

x ◦ (x ◦ y) = y ◦ (y ◦ x)

Notation. In any commutative hyper K-algebra, for all x, y ∈ H, we denote

x ∩ y = {z | z ∈ y ◦ (y ◦ x)}

Theorem 3.2. Let H be a commutative hyper K-algebra. Then we have the following

properties:

(i) x ∩ y < x and x ∩ y < y,

(ii) x ∩ y = y ∩ x,

71









(iii) x ∈ x ∩ x,

(iv) If x < y, then x ∈ x ∩ y.

for all x, y ∈ H.

Definition 3.3. A hyper K-algebra (H, ◦) is said to be positive implicative if for all x, y, z ∈ H,

(x ◦ y) ◦ z = (x ◦ z) ◦ (y ◦ z)

Theorem 3.4. Let H be a commutative and positive implicative hyper K-algebra. Then for all

x, y, z ∈ H,

(x ∩ y) ∩ z = x ∩ (y ∩ z)

Corollary 3.5. Let (H, ◦) be a commutative and positive implicative hyper K-algebra. Then

(H, ∩) is a hypersemilattice.

Example 3.6. Let H = {0, a, b} and the hyper operation ◦ is defined on H as follows:

o 0 a b

0 {0} {0,a} {0,a,b}

a {a} {0,a} {0,a,b}

b {b} {b} {0,a,b}

Then (H, ◦) is a commutative and positive implicative hyper K-algebra. Also we can see that

(H, ∩) is a hypersemilattice.

4. Relation between hyper K-algebras and superlattice

In this section we introduce the notion hypermeet ∧ on hyper K-algebras.

Definition 4.1. A hyper K-algebra (H, ◦,0) is called to be bounded, if there exist an element 1

such that x < 1, for all x ∈ H and is called complemented, if H is bounded and 1 ◦ x has a least

element with respect to <, for all x ∈ H.

We note that if H is bounded, then by (HK4) we can easily get that 1 is unique. Also if H be

complemented, then we use x' to denote min(1 ◦ x).

Notation: In any commutative hyper K-algebra, we denote

x∧ y = {z | z ∈ y◦(y◦x) s.t z < x and z <y},

for all x, y ∈ H.

Theorem 4.2. Let H be a commutative hyper K-algebra. If (x ∧ y) ∧ z = x ∧ (y ∧ z), for all x,

y, z ∈ X, then x ∧ y is a greatest lower bound of x and y.

72









Definition 4.3. Let H be a bounded commutative complemented hyper K-algebra. We say H

satisfies in conditions L, if for all x,y ∈ H,

(L1) x ∧ y ≠∅,

(L2) x' ◦ y = y' ◦ x,

(L3) (x')' = x.

Notation. In any bounded commutative hyper K-algebra with condition L, we define

x ∨ y = {z | z ∈ (x'∧ y')'}

for all x, y ∈ H.

Proposition 4.4. Let H be a bounded commutative hyper K-algebra with condition L. Then

the following hold:

(i) x < x ∨ y and y < x ∨ y,

(ii) x ∈ x ∨ x,

(iii) x ∨ y = y ∨ x,

(iv) If x < y, then y ∈ x ∨ y,

(v) If (x ∈ x ∧ y or y ∈ x ∨ y), then x < y,

for all x, y ∈ H.

Theorem 4.5. Let H be a bounded commutative hyper K-algebra with condition L. If (x∧y)∧z

= x∧(y∧z), then for all x, y, z ∈ H, (x ∨ y) ∨ z = x ∨ (y ∨ z)

Theorem 4.6. Let H be a bounded commutative hyper K-algebra with condition L and ∧ be

associative. Then x ∨ y is a lowest upper bound of x and y, for all x, y, z ∈ H.

Theorem 4.7. Let H be a bounded commutative hyper K-algebra with condition L. Then for

any x, y ∈ H, we have x ∈ x ∧ (x∨ y) and x ∈ x ∨ (x ∧ y).

Corollary 4.8. Let H be a bounded commutative hyper K-algebra with condition L and ∧ be

associative. Then (H, ∧, ∨) is a superlattice.

5. Bounded commutative hyper K-algebra of order 3 with condition L

In this section we prove that every hyper K-algebra of order 3 with condition L is a

superlattice.

Theorem 5.1. Let H = {0, a, 1} be a bounded commutative hyper K-algebra with condition L.

Then (x ∧ y) ∧ z = x ∧ (y ∧ z), for all x, y, z ∈ H.

73









Corollary 5.2. Let H = {0, a, 1} be a bounded commutative hyper K-algebra with condition L.

Then (H, ∧, ∨) is a superlattice.

In the next example we show that Theorem 5.1 is not correct for a hyper K-algebra of order

more than 3 in general.

Example 5.3. Let H = {0, a, b, 1} and the hyper operation ”◦” defined as follows:

Then (H, ◦) is a bounded commutative hyper K-algebra and satisfies in condition L. We can

see that ∧ is not associative operator.

Keyword(s): Hyper K-algebra, hypersemilattice, superlattice.

2010 AMS Classification: 03G10,06F35

Reference(s):

[1] R. A. Borzooei, A. Hasankhani and M. M. Zahedi, On hyper K-algebra, Math.Jpn., 52(2000), 113-

121.

[2] Z. Bin, X. Ying and H. S. Wei,Hypersemilattices, http://www.paper.edu.cn

[3] P. Corsini, V. Leoreanu, Applications of hyperstructure theory, Kluwer Academic Publications

(2003).

[4] Y. B. Jun, M. M. Zahedi, X. L. Xin and R. A. Borzooei, On hyper BCK-algebras, Italian Journal of

Pure and Applied Mathematics, No. 10(2000), 127-136.

[5] F. Marty, Sur une generalization de la notion degroups, 8th Congress Math. Scandinaves,

Stockholm,

(1934), 45-49.

[6] J. Mittas and M. Konstantinidou, Sur une nouvelle generation de la notion de treillis. Les

supertreillis

et certaines de leurs proprites generales, Ann. Sci. Univ. Blaise Pascal, Ser. Math., vol.25, (1989), 61-

83.

[7] T. Roodbari, L. Torkzadeh and M. M. Zahedi, Simple hyper K-algebras, Quasigroups and Related

Systems, 16 (2008), 131-140.

o 0 a b 1

0 {0, a, b,

1}

{0, a, b,

1}

{0} {0}

a {a,b,1} {0, a, b,

1}

{0,a} {0}

b {b} {b} {0, a, b,

1}

{0, a, b,

1}

1 {1} {b} {a,b,1} {0, a, b,

1}

74






1Golestan University, Department of Mathematics, Gorgan, Iran


VAGUE SOFT HYPERMODULES

T. Nozari 1

In this study, the notion of vague soft hypermodules as an extension of the notion of

vague soft hypergroups and vague soft hyperrings is introduced. Then some basic properties

of vague soft sets and homomorphisms between vague soft hypermodules are presented. Also

we studied the image and inverse image of a vague soft hypermodule under a vague soft

hypermodule homomorphism.

Keywords: Soft set, vague soft set, vague soft hypermodule.

2010 AMS Classification: 06D72 , 08A99, 20N20, 08A72.

References:

1. Ameri R, Nozari T, Fuzzy hyperalgebras, Comput. Math. Appl. 61, 2011, 149-154.

2. Ameri R, Norouzi M, H. Hedayati, Application of fuzzy sets and fuzzy soft sets

in hypermodules, RACSAM. 107, 327-338, 2013.

3. Bustince H, Burillo P, Vague sets are intuitionistic fuzzy sets, Fuzzy sets and systems. 79,

403-405, 1996.

4. Corsini P, Prolegomena of hypergroups theory, Second ed, Aviani Edotor, 1993.

5. Corsini P, Leoreanu V, Applications of hyperstructures theory, Advanced in Mathematics,

Kluwer Academic Publishers, 2003. SHORT TITLE OF THE PAPER SHOULD APPEAR

HERE 15

6. Gau W. L, Buehrer. D. J, Vague sets, IEEE Transactions on systems, Man and

Cybernetics. 23(2), 610-614, 1993.

7. Leoreanu-Fotea V, Fuzzy hypermodules,Comput. Math. Appl. 57, 466- 475, 2009.

8. Molodtsov D, Soft set theory- _rst results, Comput. Math. Appl. 37(4-5), 19-31,1999.

9. Rosenfeld A, Fuzzy groups, J. Math. Anal. Appl. 35, 512-517,1971.

10. Selvachandran G, SallehAlgebraic A. R, hyperstructures of vague soft sets associated with

hyperrings and hyperideals, Scienti_cWorldJournal. 2015.

11. Selvachandran G, SallehAlgebraic A. R, Vague soft hypergroups and vague

soft hypergroup homomorphism, Advances in Fuzzy Systems. 2014;2014:10.

doi: 10.1155/2014/758637.

12. Vougiouklis T, Hyperstructures and their representions, Hadronic Press Inc, Palm Harber,

1994.

13.. Xu W, Ma J, Wang S, Hao G, Vague soft sets and their properties, Comput.Math. Appl.

59, 787-794, 2010.

75



2 School of Mathematics, Statistic and Computer Sciences, University of Tehran, Tehran, Iran. 1,3 Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran.

E-mails: [email protected], [email protected], [email protected].

An introduction to Zero-Divisor Graph of a Commutative Multiplicative Hyperring

Z. Soltani 1 , R. Ameri 2 and Y. Talebi-Rostami 3

The purpose of this note is the study of zero-divisor graph of a commutative

multiplicative hyperrings, as a generalization of commutative rings. In this regards we consider

a commutative multiplicative hyperring (𝑅, +,∘), where (𝑅, +) is an abelian group, (𝑅,∘) is a

semihypergroup and for all 𝑎, 𝑏, 𝑐 ∈ 𝑅 , 𝑎 ∘ (𝑏 + 𝑐)⊆ (𝑎 ∘ 𝑏) + (𝑎 ∘ 𝑐) and (𝑎 + 𝑏) ∘ 𝑐 ⊆(𝑎 ∘

𝑐) + (𝑏 ∘ 𝑐). For 𝑎 ∈ 𝑅 a non-zero element 𝑏 ∈ 𝑅 is said to be a zero-divisor of a, if 0 ∈ 𝑎 ∘ 𝑏.

The set of zero-divisors of 𝑅 is denoted by 𝑍(𝑅). We associative to 𝑅 a zero-divisor graph

𝛤(𝑅), whose vertices of 𝛤(𝑅) are the elements of 𝑍(𝑅)∗ = 𝑍(𝑅)\{0} and two distinct vertices

of 𝛤(𝑅) are adjacent if they were in 𝑍(𝑅). Finally, we obtain some properties of 𝛤(𝑅) and

compare some of its properties to the zero-divisor graph of a classical commutative ring and

show that almost all properties of zero-divisor graphs of a commutative ring can be extend to

𝛤(𝑅) while 𝑅 is a strongly distributive multiplicative hyperring.

Keywords: Multiplicative hyperring, Zero-divisor graph, Fundamental relation.

AMS 2010 Classification: 20N20

References:

1. Ameri R., On the Categories of Hypergroups and Hypermodules, J. Discrete Math. Sci.

Cryptography 6, 121-132, 2003.

2. Anderson D.F., Badawi A., The total graph of a commutative ring, Journal of Algebra 320,

2706-2719, 2008.

3. Anderson D.F., Frazier A., Lauve A., Livingston P.S., The zero-divisor graph of a

commutative ring II: Lecture Notes in Pure and Appl. Math., vol. 220, Marcel Dekker, New

York, pp. 61-72, 2001.

4. Anderson D.F. , Livingston P.S., The zero-divisor graph of a commutative ring, J. Algebra,

217, 434-447,1999.

5. Beck I., Coloring of commutative rings, J. Algebra 116, 208-226, 1988.

6. Corsini P., Prolegomena of hypergroup theory, second edition Aviani, Editor, 1993.

7. Corsini P., V. Leoreanu, Applications of hyperstructure theory, Kluwer academic

publications 2003.

8. Davvaz B., Leoreanu V., Hyperrings theory and application, International Academic Press,

2007.

9. Vougiouklis T., Hyperstructures and their representations, Hardonic, press, Inc. 1994.

10. Zahedi M.M., Ameri R., On the Prime, primary and maximal subhypermodules, Italian.J.

Pure and Appl. Math., 5, 61-80, 1999.

76



1Niovi Kehayopulu(University of Athens, Department of Mathematics, 15784 Panepistimiopolis, Greece)


Some ordered hypersemigroups which enter their properties into their σ-classes

Niovi Kehayopulu 1

We are interested in ordered hypersemigroups H which enter their properties into their

σ-classes, where σ is a complete semilattice congruence on H. This gives information about

the structure of ordered hypersemigroups referring to the decomposition of these

hypersemigroups into components of the same type. We prove the following:

Theorem 1. If H is a regular, left (resp. right) regular or intra-regular ordered hypersemigroup

and σ a complete semilattice congruence on H, then the σ-class (a)σ of H is, respectively, a

regular, left (resp. right) regular or intra-regular (ordered) subsemigroup of H for every aH.

As a consequence, if H is a completely regular ordered hypersemigroup and σ a complete

semilattice congruence on H, then the σ-class (a)σ is a completely regular subsemigroup of H

for every aH.

Theorem 2. If H is a left (resp. right) quasi-regular or semisimple ordered hypersemigroup

and σ a complete semilattice congruence on H, then the σ-class (a)σ of H is, respectively so.

Theorem 3. If H is a left (resp. right) simple ordered hypersemigroup and σ a complete

semilattice congruence on H, then (a)σ is a left (resp. right) simple subsemigroup of H for

every aH.

Theorem 4. If H is a simple ordered hypersemigroup and σ a complete semilattice

congruence on H, then (a)σ is a simple subsemigroup of H for every aH.

Theorem 5. If H is an archimedean or weakly commutative ordered hypersemigroup and σ a

complete semilattice congruence on H, then (a)σ is, respectively so.

The ``-part" of the theorems above being obvious, we get characterizations of the above

mentioned types of ordered hypersemigroups via their σ-classes, σ being a complete

semilattice congruence on H.

Keywords: ordered hypersemigroup, regular, left regular, intra-regular, left quasi-regular,

semisimple, left simple, simple, archimedean, weakly commutative, complete semilattice

congruence.

2010 AMS Classification: 20M99 (06F05)

77



Study of 𝚪-hyperrings by fuzzy hyperideals with respect to a 𝒕-norm

Krisanthi Naka1, Kostaq Hila

2, Serkan Onar

3, Bayram Ali Ersoy

4

In this paper, we inquire further into the properties on some kind fuzzy hyperideals

and we study the Γ-hyperrings via 𝑇-fuzzy hyperideals. By means of the use of a triangular

norm 𝑇, we define, characterize and study the 𝑇-fuzzy left and right hyperideals, 𝑇-fuzzy

quasi-hyperideal and bi-hyperideal in Γ -hyperrings and some related properties are

investigated. We compare fuzzy hyperideal to 𝑇-fuzzy hyperideals. We have shown that Γ-

hyperring is regular if and only if intersection of any 𝑇-fuzzy right hyperideal with 𝑇-fuzzy

left hyperideal is equal to its product. We introduce the notion of 𝑇-fuzzy quasi-hyperideal

and 𝑇-fuzzy bi-hyperideal. We discuss some of its properties. We have shown that the meet of

𝑇-fuzzy right and 𝑇-fuzzy left ideal is a 𝑇-fuzzy quasi hyperideal of a Γ-hyperring. We

characterize regular Γ-hyperring with 𝑇-fuzzy quasi-hyperideal and 𝑇-fuzzy bi-hyperideal.

We also introduce the 𝑇-(𝜆, 𝜇)-fuzzy bi-hyperideals in Γ-hyperrings and investigate some of

their properties.

Keyword(s): Γ-hyperrings, 𝑡-norm, 𝑇-fuzzy (resp. left, right) hyperideal, 𝑇-fuzzy quasi(bi)-

hyperideal, 𝑇-(𝜆, 𝜇)-fuzzy bi-hyperideal.

2010 AMS Classification: 16Y99, 16D25, 20N20, 08A72.

References:

[1] Anthony, J.M., Sherwood, H. Fuzzy groups redefined, J. Math. Anal. Appl. 69, 124-130,

1979.

[2] Ameri, R., Nozari, T. A new characterization of fundamental relation on hyperrings, Int.

J. Contemp. Math. Sci. Vol. 5, no. 13-16, 721-738, 2010.

[3] Ameri, R., Shafiiyan, N. Fuzzy prime and primary hyperideals in hyperrings, Adv. Fuzzy

Math. 2, 83-99, 2007.

[4] Ameri, R., Hedayati, H., Molaee, A. On fuzzy hyperideals in Γ-hyperrings, Iranian J.

Fuzzy Syst. Vol. 6, No. 2, 47-59, 2009.

[5] Asokkumar, A., Velrajan, M. Characterizations of regular hyperrings, Ital. J. Pure Appl.

Math. No. 22, 115-124, 2007.

[6] Asokkumar, A., Velrajan, M. Hyperring of matrices over a regular hyperring, Ital. J. Pure

Appl. Math. No. 23, 113-120, 2008.

[7] Barghi, A.R. A class of hyperrings, J. Discrete Math. Sci. Cryptogr. Vol. 6, no. 2-3, 227-

233, 2003.

[8] Barnes, W.E. On the Γ-rings of Nabusawa, Pacific Journal of Mathematics, 18(3), 411-

422, 1966.

1 University of Gjirokastra, Department of Mathematics & Computer Science, Gjirokastra, Albania 2 University of Gjirokastra, Department of Mathematics & Computer Science, Gjirokastra, Albania 3 Yildiz Technical University, Department of Mathematics, Istanbul, Turkey 4 Yildiz Technical University, Department of Mathematics, Istanbul, Turkey

E-mail(s):[email protected], [email protected], [email protected], [email protected]

78




1 University of Gjirokastra, Department of Mathematics & Computer Science, Gjirokastra, Albania 2 University of Gjirokastra, Department of Mathematics & Computer Science, Gjirokastra, Albania 3 Yildiz Technical University, Department of Mathematics,Istanbul, Turkey 4 Yildiz Technical University, Department of Mathematics, Istanbul, Turkey


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80




On an algebra of fuzzy 𝒎-ary semihypergroups


2, Serkan Onar

3, Bayram Ali Ersoy

4

In this paper we deals with the fuzzy 𝑚-ary semihypergroups, fuzzy hyperideals and

homomorphism theorems on 𝑚-ary semihypergroups and fuzzy 𝑚-ary semihypergroups. We

also, introduce and study some classes of fuzzy hyperideals that of pure fuzzy, weakly pure

fuzzy hyperideals in 𝑚-ary semihypergroups and some properties of them are investigated.

We identify those 𝑚-ary semihypergroups for which every fuzzy hyperideal is idempotent.

We also characterize the 𝑚-ary semihypergroups for which every fuzzy hyperideal is weakly

pure fuzzy.

Keywords: 𝑚-ary semihypergroup, pure (weakly pure) fuzzy hyperideal, regular (weakly

regular) 𝑚-ary semihypergroups.

2010 AMS Classification: 20N15, 03E72, 20N20 .

References:

[1] Corsini, P. Prolegomena of hypergroup theory, Second edition, Aviani editor, 1993.

[2] Corsini, P., Leoreanu, V. Applications of hyperstructure theory, Advances in

Mathematics, Kluwer Academic Publishers, Dordrecht, 2003.

[3] Davvaz, B., Leoreanu, V. Binary relations on ternary semihypergroups, Commun.

Algebra 38(10), 3621-3636, 2010.

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Commun. Algebra 37, 1248-1263, 2009.

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ary hypersemigroups, Int. J. Algebra Comput. 19 (4), 567-583, 2009.

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Vol. 27(1), 37-62, 2010.



978-3-319-14762-8/ebook). x, 242 p.(2015).

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Sc. Soc. (Second Series) 23, 53-58, 2000.


E-mail(s): [email protected] , [email protected] , [email protected] , [email protected]

81








E-mails: [email protected] , [email protected], [email protected] , [email protected]

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Math. 34, 243-255, 2010.

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Math., vol. 2013, Article ID 167037, 4 pages, 2013.

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1979.

[14] Leoreanu-Fotea, V., Davvaz, B. 𝑛 -hypergroups and binary relations, European J.

Combinat., 29(5), 1207-1218, 2008.

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2718, 2009.

[16] Marty, F. Sur une generalization de la notion de group, 8th Congres Math. Scandinaves,

Stockholm, 45-49, 1934.

[17] Ostadhadi-Dehkordi, S. Semigroup derived from (Γ, N)-semihypergroups and T-functor,

Discussiones Mathematicae: General Algebra and Applications 35, 79-95, 2015.


[19] Vougiouklis, T. Hyperstructures and their representations, Hadronic Press, Florida,

1994.


82







1,2 Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran

E-maisl: [email protected] , [email protected]

On Annihilator in Pseudo BCI-algebras

Habib Harizavi 1 and Ali Bandari 2

In this paper, the concept of annihilator in a pseudo BCI-algebra is introduced and

some related properties are investigated. Some necessary and sufficient conditions for a

pseudo BCI-algebra to be semisimple are given. Moreover, it is proved that the annihilator of

a closed ideal A, denoted by *A , is the greatest closed pseudo BCI-ideal of X contained in the

BCK-part of X and satisfied * 0A A .

Keywords: pseudo BCI-algebra, pseudo BCI-ideal, annihilator, normal ideal

2010 AMS Classification: 08A99, 03B60

Acknowledgement:

Authors thank the Research Council of Shahid Chamran University of Ahvaz for its financial

support.

References:

83



The embedding of an ordered semihypergroup in terms of fuzzy sets


2, Serkan Onar

3, Bayram Ali Ersoy

4

In this paper we have investigated an embedding theorem of ordered semihypergroups in

terms of fuzzy sets. We prove that an ordered semihypergroup 𝑅 is embedded in the set 𝐹(𝑅) of all

fuzzy subsets of 𝑅, which is an poe-semigroup with the ordered relation and the multiplication and

addition defined in this paper.

Keywords: semihypergroup, ordered semihypergroup, fuzzy sets

2010 AMS Classification: 08A72, 20N20, 20N25, 06F05.

References:

[1] Bakhshi, M., Borzooei, R.A. Ordered polygroups, Ratio Math. 24, 31-40, 2013.

[2] Changphas, T., Davvaz, B. Properties of hyperideals in ordered semihypergroups, Ital. J.

Pure Appl. Math. 33, 425-432, 2014.

[3] Corsini, P. Prolegomena of Hypergroup Theory, Second edition, Aviani Editore, Italy,

1993.

[4] Corsini, P., Leoreanu, V. Applications of Hyperstructure Theory, Advances in

Mathematics, Kluwer Academic Publishers, Dordrecht, 2003.

[5] Chvalina, J. Commutative hypergroups in the sence of Marty and ordered sets,

Proceedings of the Summer School in General Algebra and Ordered Sets, Olomouck,

(1994), 19-30.

[8] Davvaz, B., Corsini, P., Changphas, T. Relationship between ordered semihypergroups

and ordered semigroups by using pseudoorder, European J. Combinatorics, 44, 208-217,

2015.

[7] Davvaz, B., Leoreanu-Fotea, V. Hyperring Theory and Applications, International

Academic Press, USA, 2007.

[8] Davvaz, B., Corsini, P., Changphas, T. Relationship between ordered semihypergroups

and ordered semigroups by using pseudoorder, European J. Combinatorics, 44, 208-

217, 2015.


2000.

[10] Davvaz, B. Fuzzy hyperideals in ternary semihyperrings, Iranian J. Fuzzy Systems, 6,

21-36, 2009.



978-3-319-14762-8/ebook). x, 242 p.(2015).

[12] Davvaz, B. A survey of fuzzy algebraic hyperstructures, Algebra Groups and

Geometries, Vol. 27(1), 37-62, 2010.


E-mails:[email protected], [email protected], [email protected], [email protected]

84





E-mails:[email protected], [email protected], [email protected], [email protected]

[13] Davvaz, B. Semihypergroup Theory, Amsterdam: Elsevier/Academic Press (ISBN 978-

0-12-809815-8/pbk; 978-0-12-809925-4/ebook). viii, 156 p. (2016).

[14] Heidari, D., Davvaz, B. On ordered hyperstructures, Politehn. Univ. Bucharest Sci. Bull.

Ser. A Appl. Math. Phys. 73(2), 85-96, 2011.

[15] Gu, Z., Tang, X. Ordered regular equivalence relations on ordered semihypergroups, J.

Algebra, 450, 384-397, 2016.

[16] Hort, D.A. A construction of hypergroups from ordered structures and their morphisms,

J. Discrete Math. Sci. Cryptogr. 6, 139-150, 2003.

[17] Kehayopulu, N., Tsingelis, M. The embedding of an ordered groupoid into a poe-

groupoid in terms of fuzzy sets, Inform. Sci. 152, 231-236, 2003.

[18] Kuroki, N. Fuzzy bi-ideals in Semigroups, Comment. Math. Univ. St. Paul. 28, 17-21,

1979.

[19] Marty, F. Sur une generalization de la notion de groupe, 8 𝑖𝑒𝑚 Congres Math.

Scandinaves, Stockholm, Sweden, 1934, 45-49.

[20] Pibaljommee, B., Davvaz, B. Characterizations of (fuzzy) bi-hyperideals in ordered

semihypergroups, J. Intell. Fuzzy Systems 28, 2141-2148, 2015.

[21] Tang, J., Davvaz, B., Luo, Y.F. Hyperfilters and fuzzy hyperfilters of ordered

semihypergroups, J. Intell. Fuzzy Systems 29(1), 75-84, 2015.


[23] Vougiouklis, T. Hyperstructures and Their Representations, Hadronic Press, Palm

Harbor, Florida, 1994.

[24] Xie, X.Y., Wang, L. An embedding theorem of hypersemigroups in terms of fuzzy sets,

2012 9th International Conference on Fuzzy Systems and Knowledge Discovery (FSKD

2012).


.

85




1 Yildiz Technical University, Department of Mathematics, Istanbul, Turkey


On δ-Primary Hyperideals of Commutative Semihyperrings

Ashraf Abumghaiseeb 1 and Bayram Ali Ersoy

1

In this work, we study the mapping 𝛿 that assigns to each hyperideal 𝐼 of the commu-

tative semihyperring 𝑅, another hyperideal 𝛿(𝐼) of the same semihyperring. Also we intro-

duced the notation of 𝛿-zero divisor of commutative semihyperring and 𝛿-semidomainlike

semihyperring which is generalization to those in the semirings. Moreover we showed that if

𝛿 be a global hyperideal expansion then 𝐼 is 𝛿-primary if and only if 𝑍𝛿(𝑅 𝐼) ⊆ 𝛿({0𝑅 𝐼⁄ })⁄ .

Keywords: Semihyperring, hyperideal, 𝛿-primary, 𝛿-semidomailike semihyperring.

2010 AMS Classification: 20N99, 13A15

References:

1. M. Shabir, N. Mehmood, P. Corsini, “Semihyperrings Characterized By Their Hyperide-

als”. Italian Journal of Pure and Applied Mathematics, accepted in March 2010.

2. R. Ameri, H. Hedayati, “On k-hyperideals of semihyperrings”, Journal of Discrete Mathe-

matical Sciences & Cryptography 10, No. 1 (2007), 41-54.

3. S. Ebrahimi Atani, Z. Ebrahimi Sarvandi and M. Shajari Kohan, “On 𝛿-Primary Ideals of

Commutative Semirings”, Romanian Journal of Mathematics and Computer Science, 2013,

Volume 3, Issue 1, P.71-81.

4. B. Davvaz, V. Leoreanu-Fote , “ Hyperring Theory and Applications”, International Aca-

demic Press. 2007.

5. P. J. Allen, “A fundamental theorem of homomorphisms for semirings”, Proc. Amer. Math.

Soc., 21(1969), 412-416.

6. Z. Dongsheng “δ-primary ideal of commutative rings”, Kyunkpook Math. Journal 41, 17-

22, 2001.

86





On the Vahlen Matrices

Mutlu Akar1

After we recall definition of Vahlen Matrices, give their some properties.

Keyword(s): Clifford Algebra, Clifford Matrix, Möbius Transformations.

2010 AMS Classification: 11E88, 15A66.

References:

1. Hile G. N. and Lounesto P., Matrix Representations of Clifford Algebras, Linear Algebra

and Its Applications, 128, 51-63, 1990.

2. Waterman P. L., Möbius Transformations in Several Dimensions, Advances in

Mathematics, 101, 87-113, 1993.

3. Ryan J., The Conformal Covariance of Huygen’s Principle-Type Integral Formulae in

Clifford Analysis, Clifford algebras and spinor structures, Math. Appl. 321, Kluwer Acad

Publ. Dordrecht, 301–310, 1995.

4. Lawson J., Clifford Algebras, Möbius Transformations, Vahlen Matrices and B-loops,

Commentationes Mathematicae Universitatis Carolinae, 51,2, 319–331, 2010.

87



1 LMAM Laboratory, University Mohamed Seddik Ben Yahia Jijel, Jijel 18000, Algeria.


On the Baireness of function spaces

Abderrahmane Bouchair 1

Let X be a topological space and α a nonempty family of compact subsets of X. Let Cα

(X) denote the space of continuous real-valued functions on X equipped with the set open

topology. A subfamily β of α is called moving off α if, for each there is with

. We say that X has the Moving Off Property (MOP) with respect to α, if every

subfamily of α which moves off α contains an infinite subfamily which has a discrete open

expansion in X.

Recently, Bouchair and Kelaiaia [1] proved that, for X paracompact q-space then Cα (X) is

Baire if and only if each point of X has a neighborhood from α. In this work we prove that if

X is first countable, then Cα (X) is a Baire space if and only if X has the Moving Off Property

with respect to α.

Keyword(s): Baire space, function space, set open topology, topological game


Reference(s):

1. A. Bouchair, S. Kelaiaia,Comparison of some set open topologies on C(X,Y), Top.

Appl. 178, 352-359, 2014.

2. R.A. McCoy, I. Ntantu, Topological properties of spaces of continuous functions,

Lecture Note in Math. 1315. Springer Verlag Germany, (1988).

88




1 Department of mathematics, University of Mohamed Seddik Ben Yahia-Jijel, Jijel, 18000 Algeria.


Global asymptotic stability of a higher order difference equation

Farida Belhannache1, Nouressadat Touafek

2

In this work, we investigate the global behavior of positive solutions of the

difference equation

𝑥𝑛+1 =𝐴+𝐵𝑥𝑛−2𝑘−1

𝐶+𝐷 ∏ 𝑥𝑛−2𝑖

𝑚𝑖𝑘𝑖=𝑙

, 𝑛 = 0,1, …

with non-negative initial conditions, the parameters 𝐴, 𝐵 are non-negative real

numbers, 𝐶, 𝐷 are positive real numbers, 𝑘, 𝑙 are non-negative fixed integers and 𝑚𝑖,

𝑖 ∈ {𝑙, … , 𝑘} are positive fixed integers such that 𝑙 ≤ 𝑘.

Keyword(s): Difference equation, global behavior, oscillatory, boundedness


Reference(s):

1. Abo-Zeid R., Global behavior of a higher order difference equation, Math. Solvaca.,

64, 4, 931-940, 2014.

2. Belhannache F, Touafek N, Abo-Zeid R, Dynamics of a third-order rational difference

equation, Bull. Math. Soc. Sci. Math. Roumanie., 59, 1, 13-22, 2016.

3. Elsayed E. M, On the dynamics of a higher-order rational recursive sequence,

Commun. Math. Anal., 12, 117-133, 2012.

89



1Department of Mathematics, University of Annaba, Algeria 2Department of Mathematics University of Souk Ahras, Algeria

E-mails: [email protected] , [email protected].

Weak -favorability of C(X) with a set open topology

Kelaiaia Smail1 and Harkat Lamia

2

Let C(X) be the set of all continuous real valued functions we endow it with a set open

topology with the help of a family of -compact subsets of X. We use a topological game to

study the weak -favorability of C(X).

Keywords: Function spaces, set-open topologies, uniform topologies, topological games,

-favorability, -compact.

2010 AMS Classification: 54C35.

References:

1.A. Bouchair and S. Kelaiaia, "Application des jeux topologiques à l'étude de C(X) muni

d’une topologie set-open", Revue des Sciences etTechnologies. Univ. Mentouri. Constantine.

A-No 20.17-20. (2003).

2. G. Choquet, "Lectures in analysis," Benjamin, New York, Amsterdam, 1969.

3. R. Egelking, "General Topology," Polish scientifc publishing, 1977.

4. G. Gruenhage, "Games, covering properties and Eberlein compacts," Top.Appl. 23(1986),

291-297.

5. S. Kelaiaia, "On a completeness property of C(X)," Int. J. Appl. Math.6(2001), 287-291.

6. S. E. Nokhrin and A. V. Osipov, "On the Coicidence of Set-Open and Uniform

Topologies," Proc. Steklov. Inst. Suppl. 3(2009), 184-191.

7. R. A. McCoy and I. Ntantu, "Topological Properties of Spaces of Continuous Functions,"

Lecture Note, Springer-Verlag, Berlin, (1988).

8. R. A. McCoy and I. Ntantu, "Countability properties of function spaces with set-open

topology," Top. Proc. 10(1985), 329-345.

9. R. telgarsky, "Topological games on the 50th anniversary of the BanachMazur game,"

Topology Proc. 10, 329-345. (1987).

10. J.C. Oxtoby, "The Banach-Mazur games and Banach category theorem," Contribution to

the theory of games, Vl. III Annals of Math.Studies 39. Princeton UniversityPress. Princeton,

(1957).

90



1,2Department of Mathematics and Applications, University of Mohaghegh Ardabili, Ardabil, Iran.


Curves on Lightlike Cone in Minkowski Space

Nemat Abazari1, Alireza Sedaghatdoost

2

In the Minkowski space 𝐸1𝑛, the set of all lightlike vectors is called lightlike cone and

it is denoted by 𝑄𝑛−1 . In this paper we use Frenet orthonormal frame and asymptotic

orthonormal frame for study of curves on the lightlike cone 𝑄𝑛−1 in Minkowski space 𝐸1𝑛. We

study all lightlike and spacelike curves in𝑄𝑛−1. We classify all curves with constant cone

curvature in 𝑄4, 𝑄5 and 𝑄6. Also we give some relation between Frenet curvature and cone

curvature functions for a curve in 𝑄3.

Keyword(s): Asymptotic frame, cone curvature, lightlike cone, spacelike curve.


Reference(s):

1. D.N. Kupeli , On null submanifolds in spacetime,Geometriae Dedicata 23 (1987), 33-51.

2. H. Liu, Curves in the lightlike cone. Contrib. Algebr. Geom. 45 (2004), 291-303.

3. H. Liu, Q. Meng, Represeentation formulas of curves in a two- and threeDimensional

lightlike cone, Results in Math. 59 (2011), 437-451.

4. R. López, On differential geometry of curves and surfaces in LorentzianMinkowski

space, International Electronic Journal of Geometry. 7 (2014), 44-107.

5. B. O’Neill, Semi-Riemannian Geometry, Academic Press, NewYork 1983.

6. M. P. Torgaˇsev, E. Sućurović, ˇ W-Curves in Minkowksi space-time, Novi SaJ.

Math. Vol 32, No 2, 2002, 55-65.

7. S. Yilmaz, M. Turgut, On the differential geometry of the curves in Minkowski

space-time I, Int. J. Contemp. Math. Sciences, Vol. 3, 2008, no. 27, 1343-1349.

91




1,2Yildiz Technical University, Department of Mathematics, Istanbul, Turkey


*This research is supported by Yildiz Technical University Scientific Research Projects Coordination

Department Project Number: 2016-01-03-DOP01.

A class of LCD codes from group rings*

Mehmet E. Koroglu1, Bayram A. Ersoy

1

Linear codes with complementary duals (abbreviated LCD) are linear codes that they

trivially intersect with their duals [1]. Finding new LCD code families are of great importance

due to their wide range of applications. Unit derived group ring codes were given by Hurley

and Hurley in [2]. Group rings are a rich source of unit elements. So it is possible to obtain

many new code parameters. In this work, we provide a necessary and sufficient condition for

the unit derived group ring codes to be LCD.

Keywords: Linear codes, LCD codes, group rings

2010 AMS Classification: 94B05, 94B60, 08A99

References:

1. Massey, J. L. Linear codes with complementary duals, Discrete Math. 106, 337-342, 1992.

2. Hurley, P. and Hurley, T. Codes from zero-divisors and units in group rings, International

Journal of Information and Coding Theory 1, 57-87, 2009.

92



1Balikesir University, Department of Mathematics, Balikesir, 10145, Turkey 2Balikesir University, Department of Mathematics, Balikesir, 10145, Turkey 3Balikesir University, Department of Mathematics, Balikesir, 10145, Turkey


New Blocking Cryptography Models

Sümeyra UÇAR1, Nihal TAŞ

2, Nihal Yılmaz Özgür

3

In this talk we present two new coding and decoding methods using Fibonacci Q

matrices and R-matrices. Since our methods study with small numbers, we obtain quite easy

methods than the known methods in literature.

Keyword(s): Coding algorithm / decoding algorithm / Fibonacci Q matrix / R matrix

2010 AMS Classification: 68P30, 11B39, 1B37.

Reference(s):

1. Bruggles I. D., Hoggatt V. E. Jr., A Primer for the Fibonacci numbers-Part IV. Fibonacci

Q. 1 (4), 65-71, 1963.

2. Koshy, T., Fibonacci and Lucas numbers with applications, New York, NY: JohnWiley and

Sons, 2001.

3. Prasad, B., Coding theory on Lucas p numbers, Discrete Mathematics, Algorithms and

Applications 8 (4), 17 pages, 2016.

4. Stakhov, A. P., Fibonacci matrices, a generalization of the Cassini formula and a new

coding theory, Chaos, Solitons Fractals 30 (1), 56-66, 2006.

5. Taş, N., Uçar, S. and Özgür, N. Y., Pell coding and Pell decoding methods with some

applications, arXiv:1706.04377 [math.NT].

6. Taş, N., Uçar, S., Özgür, N. Y. and Kaymak, Ö. Ö., A new coding/decoding algorithm

using Fibonacci numbers, submitted for publication.

7. Uçar, S., Taş, N. and Özgür, N. Y., A new cryptography model via Fibonacci and Lucas

numbers, submitted for publication.

93





A New Coding Theory with Generalized Pell (p,i) – Numbers

Nihal Taş1, Sümeyra Uçar

2, Nihal Yılmaz Özgür

3

Recently, it has been introduced a new coding algorithm, called blocking algorithm,

using Fibonacci (resp. Lucas) numbers and a blocking method. In this study, we develop a

new coding and decoding method using the generalized Pell (p,i) – numbers. We give an

application of generalized Pell (p,i) – numbers to blocking algorithm.

Keyword(s): Coding theory / decoding theory / generalized Pell (p,i) – numbers / blocking

algorithm

2010 AMS Classification: 68P30, 14G50, 11T71, 11B39.

Reference(s):

1. Kılıç, E., The generalized Pell (p,i)-numbers and their Binet formulas, combinatorial

representations, sums, Chaos, Solitons Fractals, 40, 2047-2063, 2009.

2. Koshy, T., Pell and Pell-Lucas numbers with applications, Springer, Berlin, 2014.

3. Prasad, B., Coding theory on Lucas p numbers, Discrete Mathematics, Algorithms and

Applications 8 (4), 17 pages, 2016.

4. Stakhov, A. P., Fibonacci matrices, a generalization of the Cassini formula and a new

coding theory, Chaos, Solitons Fractals 30 (1), 56-66, 2006.

5. Taş, N., Uçar, S. and Özgür, N. Y., Pell coding and Pell decoding methods with some

applications, arXiv:1706.04377 [math.NT].

6. Taş, N., Uçar, S., Özgür, N. Y. and Kaymak, Ö. Ö., A new coding/decoding algorithm

using Fibonacci numbers, submitted for publication.

7. Uçar, S., Taş, N. and Özgür, N. Y., A new cryptography model via Fibonacci and Lucas

numbers, submitted for publication.

94



1Nevşehir Hacı Bektaş Veli University, Department of Mathematics, Nevşehir Turkey 2Nevşehir Hacı Bektaş Veli University, Department of Mathematics, Nevşehir Turkey 3Nevşehir Hacı Bektaş Veli University, Department of Mathematics, Nevşehir Turkey

E-mail(s):[email protected], [email protected], [email protected]

Some Problems in Spectral Graph Theory

Sezer Sorgun1, Hakan Küçük

2, Hatice Topcu

3

Spectral Graph Theory is the studies of Linear Algebra and Graph Theory mix

together. In this study, we usually find the relations between the eigenvalues of popular

matrices and the graph parameters. In this talk, we present some problems related to energy

problems, isomorphism etc. in the theory.

Keyword(s): Graph, Isomorphism, Energy, Graph Matrices, Eigenvalue


Reference(s):

1. Haemers W., Bouwer A., Spectra of Graphs, Springer, 2010.

2. Haemers W., Dam E. R., Which Graphs are Determined by Their Spectrum?, Linear

Algebra Appl., 373, 241-272, 2003.

3. Li X., Shi Y., Gutman I., Graph Energy, Springer, New York, 2012.

4. Das K.C., Sorgun S., Gutman I., On Randić energy, MATCH Comm.Math.Comput. Chem.,

73, 81-92, 2015.

5. Das K.C., Sorgun S., On Randić energy of graphs, MATCH Comm.Math.Comput. Chem.,

72, 227-238, 2014.

6. Zhang X., Zhang H., Some Graphs Determined by Their Spectra, Linear Algebra Appl.

431,1443-1454, 2009.

7. Sorgun S., Topcu H., On the spectral characterization of kite graphs_ J. Algebra Comb.

Discrete Appl., 3 (2), 81–90, 2016.

8. Topcu H., Sorgun S., Haemers W., On the spectral characterization of pineapple graphs,

Linear Algebra and its Applications 507, 267–273, 2016.

95



1Nevşehir Hacı Bektaş Veli University, Department of Mathematics, Nevşehir Turkey 2Nevşehir Hacı Bektaş Veli University, Department of Mathematics, Nevşehir Turkey 3Nevşehir Hacı Bektaş Veli University, Department of Mathematics, Nevşehir Turkey


On Trees Which Have Four Non-Zero Randić Eigenvalues

Hakan Küçük1, Sezer Sorgun

2, Hatice Topcu

3

A popular and important research field is to investigate the characterization of the

connected graphs with special and distinct eigenvalues. It is an interplay between

combinatorics and linear algebra. Moreover, Randić Matrix and Randić Energy studies in

Spectral Graph Theory are essential. In this presentation we give some basic information

about Randić Matrix, then we present our observations and conclusions about trees which

have four non-zero Randić eigenvalues.

Keyword(s): Graph, Matrices, Randić Matrix, Randić Eigenvalues, Trees


Reference(s):

1. Diestel R., Graph Theory, volume 173 of Graduate Texts in Mathematics. Springer-

Verlag, Heidelberg, 2010.

2. Chung F., Spectral Graph Theory, American Mathematical Society,National Science

Foundation, 1997.

3. Gu R. , Huang F. , Li X. , General Randić Matrix and General Randić Energy,

Transactions on Combinatorics, 3 (3) , 21-33, 2014.

4. Bozkurt Ş.B., Güngör A.D., Gutman I. , Çevik A. S., Randić Matrix and Randi¢

Energy, MATCH Commun. Math. Comput. Chem., 64 , 239-250, 2014.

5. Li X. , Wang J. , Randić Matrix and Randić Eigenvalues, MATCH Commun. Math.

Comput. Chem. 73, 73-80, 2015.

6. Dam E.R. , Graphs with few eigenvalues: An interplay between combinatorics and

algebra, PhD Thesis, Tilburg University, 1996.

96






One-Parameter Planar Motions in Generalized Complex Number Plane

Nurten (Bayrak) Gürses1 and Salim Yüce

1

The generalized complex numbers have the form

2, ( , ) where , ( , ).z x Jy x y J i q p p q

By taking 2 ; 0J p q and p , generalized complex number system can be

presented as follows:

2{ : , , }x Jy x y J p p .

p is called p -complex plane. Moreover, the set J is defined

2: , , , { 1,0,1}J x Jy x y J p p such that J ⊂ p . For p <0, p is called

elliptical complex, for p =0, p is called parabolic complex, and for p >0, p is called

hyperbolic complex number systems.

In this study, we firstly give the basic notations of the p -complex plane p . Then, we

introduce the one-parameter planar motions in p -complex plane J such that J ⊂ p .

These motions correspond the one-parameter motions in affine Cayley-Klein planes. We

examine this motion theory with aspects of complex motions. Besides, we discuss the

relations between absolute, relative, sliding velocities (accelerations) and pole curves under

the motions J / J .

Keywords: Generalized complex number plane, complex-type numbers, one-parameter

planar motion, kinematics.

2010 AMS Classification: 53A17, 53A35

References:

[1] N. (Bayrak) Gürses, S. Yüce, One-Parameter Planar Motions in Affine Cayley-Klein

Planes. European Journal of Pure and Applied Mathematics, 7 no. 3(2014), 335–342.

[2] W. Blaschke and H. R.Müller, Ebene Kinematik. Verlag Oldenbourg, München,1956.

[3] P. Fjelstad, Extending special relativity via the perplex numbers. Amer. J. Phys. 54 no.5

(1986), 416–422.

97





[4] D. Alfsmann, On families of 2N dimensional Hypercomplex Algebras suitable for digital

signal Processing. Proc. EURASIP 14th European Signal Processing Conference(EUSIPCO

2006), Florence, Italy, 2006.

[5] I.M. Yaglom, Complex numbers in geometry, Academic Press, New York, 1968.

[6] S. Yüce and N. Kuruoğlu, One-Parameter Plane Hyperbolic Motions. Adv. appl. Clifford

alg. 18 (2008), 279–285.

[7] P. Fjelstad and S.G. Gal, n-dimensional hyperbolic complex numbers. Adv. Appl. Clifford

Algebr. 8 no.1 (1998), 47–68.

[8] P. Fjelstad and S.G. Gal, Two-dimensional geometries, topologies, trigonometries and

physics generated by complex-type numbers. Adv. Appl. Clifford Algebra 11 no. 1 (2001)

81–107.

[9] D. Rochon and M. Shapiro, On algebraic properties of bicomplex and hyperbolic numbers.

Anal. Univ. Oradea, fasc. math. 11 (2004), 71–110.

[10] F. Catoni, R. Cannata, V. Catoni and P. Zampetti, Hyperbolic Trigonometry in two-

dimensional space-time geometry. N. Cim. B 118 B (2003), 475491.

[11] E. Study, Geometrie der Dynamen. Verlag Teubner, Leipzig, 1903.

[12] F. M. Dimentberg, The Screw Calculus and Its Applications in Mechanics, Foreign

Technology Division translation FTD-HT-23-1632-67, (1965).

[13] F. M. Dimentberg, The method of screws and calculus of screws applied to the theory of

three dimensional mechanisms. Adv. in Mech. 1 no. 3-4 (1978), 91–106.

[14] Ö. Köse, Kinematic differential geometry of a rigid body in spatial motion using

dual vector calculus: Part-I. Applied Mathematics and Computation 183 no. 1 (2006), 17–29.

[15] G.R. Veldkamp, On the use of dual numbers, vectors and matrices in instantaneous,

spatial kinematics. Mechanism and Machine Theory 11 no. 2 (1976), 141–156.

[16] A. A. Harkin and J. B. Harkin, Geometry of Generalized Complex Numbers.

Mathematics Magazine, 77 no. 2 (2014).

[17] F. Catoni, D. Boccaletti, , R. Cannata, V. Catoni, E. Nichelatti and P. Zampetti, The

mathematics of Minkowski space-time and an introduction to commutative hypercomplex

numbers. Birkh auser Verlag, Basel, 2008.

[18] I. M. Yaglom, A simple non-Euclidean geometry and its Physical Basis. Springer-Verlag,

New York, 1979.

[19] E. Pennestri and R. Stefanelli, Linear Algebra and Numerical Algorithms Using Dual

Numbers. Multibody System Dynamics, 18 no. 3 (2007), 323–344.

[20] G. Sobczyk, The Hyperbolic Number Plane. The College Math. J. 26 no. 4 (1995), 268–

280.

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98





[23] H. Es, Motions and Nine Different Geometry. PhD Thesis, Ankara University Graduate

School of Natural and Applied Sciences, 2003.

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107 no. 3 (2000), 219–237.

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framework. http://arxiv.org/pdf/physics/9702030v1.pdf., (1997).

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Relativity to Undergraduates, AAPT Summer Meeting, Syrauce University, NY, July 20-

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164–167.

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Geometry: Methods and Applications, 5 (072) (2009).

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Actions of SL2 (R). Imperial College Press, London, 2012.

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Cayley-Klein spaces and kinematic groups. Hadronic J. 19 no. 4 (1996), 407–435.

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463–469.

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99



1Nevşehir Hacı Bektaş Veli University, Department of Mathematics, Nevsehir, Turkey,

2Ankara University, Department of Mathematics, ,Ankara, Turkey,


A New Approach to Motions and Surfaces with Zero Curvatures in Lorentz 3-Space

Esma DEMİR ÇETİN 1 and Yusuf YAYLI

2

In this work we search for the surfaces with zero curvatures in Lorentz 3-space, whose

generating curve is a graph of a polynomial under homothetic motion groups. We study with

the generating curves α (s) = (f(s),0,g(s)), α (s) = (f(s),g(s),0), α (s) =(f(s),g(s),f(s)) depending

on the casual character of the axis. (Timelike axis, spacelike axis, lightlike axis respectively.)

First of all we see that the degree of the polynomials must be equal for zero curvatures. We

show that, distinct from the helicoidal motion groups, these surfaces generated by graph of

polynomials don’t have to be ruled surfaces for zero curvatures.

Keywords: Gauss curvature, mean curvature, umbilic points, Lorentz space, Homothetic

motion

2010 AMS Classification: 53B30, 53C50.

References:

100



1 Balıkesir University, Department of Mathematics, Balıkesir, Turkey, 2 Balıkesir University, Department of Mathematics, Balıkesir, Turkey,


Chen-Ricci and Wintgen Inequalities for Statistical Submanifolds of Quasi-Constant

Curvature

Hülya Aytimur 1, Cihan Özgür

2

We define statistical manifold of quasi-constant curvature and give an example. We

find Chen-Ricci inequalities, generalized Wintgen inequality for submanifolds in a statistical

manifold of quasi-constant curvature.

Keywords: Statistical submanifold \ Chen-Ricci İnequalities \ Wintgen İnequality \ quasi-constant

curvature

2010 AMS Classification: 53C40, 53B05, 53B15, 53C05, 53A40

References:

1. S. Amari, Differential-Geometrical Methods in Statistics, Springer-Verlag, 1985.

2. M. E. Aydın, A. Mihai, I. Mihai, Some Inequalities on Submanifolds in Statistical

Manifolds of Constant Curvature, Filomat 29 (2015), no. 3, 465-477.

3. M. E. Aydın, A. Mihai, I. Mihai, Generalized Wintgen inequality for statistical

submanifolds in statistical manifolds of constant curvature, Bull. Math. Sci. (2016).

doi:10.1007/s13373-016-0086-1

4. P. W. Vos, Fundamental equations for statistical submanifolds with applications to the

Bartlett correction, Ann. Inst. Statist. Math. 41 (1989), no. 3, 429-450.

5. C. Özgür, B. Y. Chen inequalities for submanifolds of a Riemanian manifold of quasi-

constant curvature, Turk. J. Math. 35 (2011) 501-509.

6. B.-Y. Chen, Geometry of submanifolds. Pure and Applied Mathematics, No. 22. Marcel

Dekker, Inc., New York, 1973.

7. H. Furuhata, Hypersurfaces in statistical manifolds, Differential Geom. Appl. 27 (2009),

no. 3, 420-429.

101



1Yildiz Technical University, Department of Mathematics,Istanbul,Turkey

2Yildiz Technical University, Department of Mathematics,Istanbul,Turkey


One-Parameter Homothetic Motion on the Galilean Plane

Mücahit Akbıyık1, Salim Yüce

2

In this paper, we will define one-parameter homothetic motion on the Galilean Plane.

The velocities, accelerations and pole points of the motion will be analysed.

Keywords: Galilean(Isotropic) Plane, Kinematics


References:

1. Edmund Taylor Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid

Bodies, Cambridge University Press. Chapter 1, 1904.

2. Joseph Stiles Beggs, Kinematics, Taylor and Francis. p. 1, 1983.

3. Thomas Wallace Wright, Elements of Mechanics Including Kinematics, Kinetics and

Statics. E and FN Spon. Chapter 1, 1896.

4. A. Biewener, Animal Locomotion, Oxford University Press, 2003.

5. Blaschke, W., and Müller, H.R., Ebene Kinematik, Verlag Oldenbourg, München, 1959.

6. A. A. Ergin, On the one-parameter Lorentzian motion,Comm. Fac. Sci. Univ. Ankara,

Series A 40, 59-66, 1991.

7. Akar, M., Yüce, S. and Kuruoglu, N, One parameter planar motion in the Galilean plane,

International Electronic Journal of Geometry, 6(1), 79-88, 2013.

8. Hacisalihoğlu, H., On the rolling of one curve or surface upon another, Proceedings of the

Royal Irish Academy. Section A: Mathematical and Physical Sciences, Vol. 71, (1971), pp.

13-17.

9. Tutar, A., and Kuruoglu, N, On the one-parameter homothetic motions on the Lorentzian

plane, Bulletin of Pure & Applied Sciences, Vol. 18E, No.2 , 333-340, 1999.

10. Yaglom, I.M., A simple non-Euclidean Geometry and its Physical Basis, Springer-Verlag,

New York, 1979.

11. O. Röschel, Die Geometrie des Galileischen Raumes, Habilitationsschrift, Institut für

Math. und Angew. Geometrie, Leoben, 1984.

12. Helzer, G., Special relativity with acceleration, Mathematical Association of America.

The American Mathematical Monthly, Vol. 107, No. 3, 219-237, 2000.

102


24-27 July 2017, Istanbul-Turkey

1Yildiz Technical University, Department of Mathematics, Istanbul, Turkey.

2Yildiz Technical University, Department of Mathematical Engineering, Istanbul, Turkey.

Emails: [email protected], [email protected], [email protected]

Comparison of encryption and decryption algorithms through various

approaches

Murat Sari1, Meliha İpek Bulut

2, İlter Beren Kanpak

2

The aim of this paper is to produce computer codes of fundamental encryption and

decryption algorithms in a comparison way through various approaches such as Substitution,

Affine, Vigenere, Caesar and RSA. Performances of the corresponding approaches have been

deeply investigated by comparing the usage RAM and CPU times. The codes for the

encryption and decryption algorithms are written in C#. General summary of cryptography

has also been presented. Effects and performances of computer codes of encryption and

decryption algorithms for each one of the methods Affine, Vigenere, Caesar, RSA,

Substitution have been compared. For the encryption algorithms, the RSA has been seen to be

the best for performance time. For the encryption algorithms, the Affine has been seen to be

the best for CPU. For the decryption algorithms, the Substitution has been seen to be the best

for performance time. Note also that, for the decryption algorithms, the RSA has been seen to

be the best for CPU. Increasing the encryption speed by changing the encryption algorithms is

an open problem. Future work can focus on this problem.

Keywords: Cryptology, Encryption algorithms, Decryption algorithms, Substitution, Affine,

Vigenère, Caesar, RSA.

2010 AMS Classification: 14G50, 11T71

References:

1. Singh S., The Code Book, September 1999.

2. Purnama B. and Hetty R.A.H., A New Modified Caesar Cipher Cryptography Method with

Legible Ciphertext from a Message to be Encrypted, Procedia Computer Science, 59, 195-204

2015.

3. Simmons G.J., Vigenère Cipher Cryptology, Britannica, April 2017.

4. Niederreiter H., Winterhof A., Applied Number Theory, Springer, 2015

5. Kaufmann M., Computer and Information Security Handbook, 2009.

6. Paar C. and Pelzl J., Understanding Cryptography, Springer, 2010.

7. Kartalopoulos S.V., Next Generation Intelligent Optical Networks, 191 C, Springer 2008.

8. Daras N.J. (ed.), Applications of Mathematics and Informatics in Science and Engineering,

Springer Optimization and Its Applications 91, Springer, 2014.

103






1 Nevşehir HBV University, Department of Mathematics, Nevşehir,TURKEY

2Ankara University, Mathematics Department, Ankara, TURKEY


Surfaces with Constant Slope and Tubular Surfaces

Çağla RAMİS 1 and Yusuf YAYLI 2

Tubular surfaces can be characterized as a subfamily of canal surfaces with the

constant radius. In this study, we develop the endowed reduced definition of tubular surface

and give the new general parameterization by non perpendicular circle along the base curve.

Moreover, the advantage of new characterization is to yield a tubular surface without singular

points. In accordance with this purpose, we also focus to eliminate singular points by the

location of circle which moves along the base curve of surface. Mathematical description of

these surfaces enables the relation with constant slope surfaces and the creation of their

modeling on computer.

Keywords: Canal surface, tubular surface, surface with constant slope, singularity.


References:

104





New Contributions to Fixed-Circle Results on S-Metric Spaces

Ufuk ÇELİK1, Nihal YILMAZ ÖZGÜR

2, Nihal TAŞ

3

Recently, it has been given some fixed-circle theorems on metric and S-metric spaces.

In this talk, we present some new fixed-circle theorems on S-metric spaces. We give new

examples of S-metrics and investigate some relationships between circles on metric and S-

metric spaces. Then we investigate some existence and uniqueness conditions for fixed circles

of self-mappings.

Keyword(s): Fixed circle / fixed-circle theorem / existence theorem / uniqueness theorem / S-

metric

2010 AMS Classification: Primary: 47H10, Secondary: 54H25, 55M20, 37E10.

Reference(s):

1. Gupta, A., Cyclic Contraction on S-Metric Space, International Journal of Analysis

and Applications 3, no.2, 119-130, 2013.

2. Hieu, N. T., Ly, N.T. and Dung, N.V. A Generalization of Ciric Quasi-Contractions

for Maps on S-Metric Spaces, Thai Journal of Mathematics 13, no.2, 369-380, 2015.

3. Özgür, N.Y. and Taş, N. Some fixed point theorems on S-metric spaces, Mat. Vesnik

69, no.1, 39-52, 2017.

4. Özgür, N.Y. and Taş, N. Some new contractive mappings on S-metric spaces and their

relationships with the mapping (S25), Math. Sci. 11, no.7, doi:10.1007/s40096-016-0199-4,

2017.

5. Özgür, N.Y. and Taş, N. Some fixed circle theorems on metric spaces,

arXiv:1703.00771v1 [math.MG].

6. Özgür, N.Y. and Taş, N. Some fixed circle theorems on S-metric spaces with a geometric

viewpoint, submitted for publication.

7. Ögür, N.Y. and Taş, N. Some Generalizations of Fixed Point Theorems on S-Metric

Spaces, Essays in Mathematics and Its Applications in Honor of Vladimir Arnold, New

York, Springer, 2016.

8. Sedghi, S., Shobe N. and Aliouche, A. A Generalization of Fixed Point Theorems in

S-Metric Spaces, Mat. Vesnik 64, no.3, 258-266, 2012.

105



1 Kilis 7 Aralık University, Department of Mathematics, , Kilis, TURKEY 1 2 Eskisehir Osmangazi University, Department of Mathematics-Computer ,Eskişehir, TURKEY 2 3 Eskisehir Osmangazi University, Department of Mathematics-Computer ,Eskişehir, TURKEY 3


On The Parallel Ruled Surfaces With B-Darboux Frame

Mustafa Dede1, Hatice TOZAK

2, Cumali Ekici

3

In this paper, the parallel ruled surfaces with B-Darboux frame are introduced in

Euclidean 3-space. Then some characteristic properties of the parallel ruled surfaces with B-

Darboux frame such as developability, striction point and distribution parameter are given in

E³.

Keyword(s): Parallel ruled surface/ Darboux frame/ Surfaces.

2010 AMS Classification: 53A05, 53A15,53R25

Reference(s):

1. Çöken A. C., Çiftçi Ü. and Ekici C., On parallel timelike ruled surfaces with timelike

rulings, Kuwait J. Sci. Engrg., 35(1), 21-31, 2008.

2. Gray A., Salamon S. and Abbena E., Modern differential geometry of curves and surfaces

with Mathematica, Chapman and Hall/CRC, 2006.

3. O'Neill B., Elementary differential geometry, Academic Press Inc, New York, 1996.

4. Klok F., Two moving coordinate frames for sweeping along a 3D trajectory, Comput.

Aided Geom. Des., 3(3), 217-229, 1986.

5. Darboux G., Leçons sur la theorie generale des surfaces I-II-III-IV., Gauthier-Villars, Paris,

1896.

6. Şentürk G. Y. and Yüce S., Characteristic properties of the ruled surface with Darboux

frame in E³, Kuwait J. Sci., 42(2), 14-33, 2015.

7. Hacısalihoğlu H. H., Diferensiyel geometri, İnönü Üniv. Fen Edebiyat Fak. Yayınları, 2,

1983.

8. Shin H., Yoo S. K., Cho S. K. and Chung W. H., Directional offset of a spatial curve for

practical engineering design, ICCSA, 3, 711-720, 2003.

9. Hoschek J., Integral invarianten von regelflachen, Arch. Math, XXIV, 1973.

10. Bloomenthal J., Calculation of reference frames along a space curve, Graphics Gems,

Academic Press Professional, Inc., San Diego, CA., 1990.

11. Dede M., Ekici C. and Görgülü A., Directional B-Darboux frame along a space curve.

IJARCSSE, 5, 775-780, 2015.

12. Dede M., Ekici C. and Tozak H., Directional tubular surfaces, Int. J. Algebra, 9(12), 527-

535, 2015.

13. Yüksel N., The ruled surfaces according to Bishop frame in Minkowski space, Abstr.

Appl. Anal., http://dx.doi.org/10.1155/2013/810640, 2013, 1-5, 2013.

14. Carmo P.M., Differential geometry of curves and surfaces, Prentice-Hall, Englewood

Cliffs, New York, 1976.

106



1 Kilis 7 Aralık University, Department of Mathematics, , Kilis, TURKEY 1 2 Eskisehir Osmangazi University, Department of Mathematics-Computer ,Eskişehir, TURKEY 2 3 Eskisehir Osmangazi University, Department of Mathematics-Computer ,Eskişehir, TURKEY 3


15. Bishop R.L., There is more than one way to frame a curve, Am. Math. Mon., 82, 246-251,

1975.

16. Coquillart S., Computing offsets of B-spline curves, Comput. Aided Des., 19, 305-309,

1987.

17. Ravani T. and Ku S., Bertrand offsets of ruled surface and developable surface, Comput.

Aided Geom. Des., 23(2), 145-152, 1991.

18. Maekawa T., Patrikalakis N.M., Sakkalis T. and Yu G., Analysis and applications of pipe

surfaces, Comput. Aided Geom. Des., 15, 437-458, 1988.

19. Hlavaty V., Differentielle linien geometrie, Uitg P. Noorfhoff, Groningen, 1945.

20. Wang W., Jüttler B., Zheng D. and Liu Y., Computation of rotation minimizing frames,

ACM Trans. Graph.27, 1-18, 2008.

21. Kühnel W., Differential geometry, curves-surfaces-manifolds, Am. Math. Soc., 2002.

22. Ünlütürk Y., Çimdiker M. and Ekici C., Characteristic properties of the parallel ruled

surfaces with Darboux frame in Euclidean 3-space, Commun. Math. Model. Appl., 1(1), 26-

43, 2016.

23. Savcı Z., Görgülü A. and Ekici C., On Meusnier theorem for parallel surfaces, Commun.

Fac. Sci. Univ. Ank. S. A1 Math. Stat., 66(1),187-198, 2017.

107



1Istanbul University,Department of Mathematical Education,Istanbul, Turkey 2 Istanbul University,Department of Mathematical Education,Istanbul, Turkey


An ANFIS Perspective for the diagnosis of type II diabetes

Murat Kirişci1, M. Ubeydullah Saka

2

An adaptive network is a multilayer feed forward network in which each node

performs a particular function (node function) on incoming signals as well as asset of

parameters pertaining to this node. Fuzzy inference systems are the fuzzy rule based systems

which consists of a rule base, database, decision making unit, fuzzification interface and a

defuzzification interface. By embedding the fuzzy inference system into the framework of

adaptive networks, a new architecture namely Adaptive neuro fuzzy inference system

(ANFIS) is formed which combines the advantages of neural networks and fuzzy theoretic

approaches.

In this study ANFIS is presented for the diagnosis of diabetes diseases. The ANFIS

classifier is used to diagnose diabetes disease when six features defining diabetes indications

are used as inputs. The proposed ANFIS model is then evaluated and its performance is

reported. We are able to achieve significant improvement in accuracy by applying the ANFIS

model. Finally, some conclusions are drawn concerning the impacts of features on the

diagnosis of diabetes disease.

Keyword(s): diabetes, fuzzy logic, adaptive neuro-fuzzy inference system(ANFIS)

2010 AMS Classification: 68T05, 92C50, 03E72

Reference(s):


2. Zadeh, L.A., Biological application of the theory of fuzzy sets and systems, The

proceedings of an International Symposium on Biocybernetics of the Central Nervous

System, 199-206, 1969.

3. Allahverdi, N., Design ıf Fuzzy expert systems and its applications in some medical areas,

International Journal of Applied Mathematics, Electronics and Computers, 2(1), 1-8, 2014.

4. Torres A. & Nieto J. J. Fuzzy Logic in Medicine and Bioinformatics, Journal of

Biomedicine and Biotechnology, Vol. 2006, 1–7, 2006.

5. Mahfouf M., Abbod M. F. & Linkens D. A. A survey of fuzzy logic monitoring and

control utilisation in medicine, Artificial Intelligence in Medicine, 21(1–3), 27–42, 2001.

6. Polat, K., & Gunes, S., An expert system approach based on principal component analysis

and adaptive neuro-fuzzy inference system to diagnosis of diabetes disease. Digital Signal

Processing, 17(4), 702–710, 2007.

7. Mohamed, E. I., Linderm, R., Perriello, G., Di Daniele, N., Poppl, S. J., & De Lorenzo, A.,

Predicting type 2 diabetes using an electronic nose-base artificial neural network analysis.

Diabetes Nutrition & Metabolism, 15(4),

215–221, 2002.

108





Transitive Operator Algebras and Hyperinvariant Subspaces

Elif Demir1

In this work, I deal with the transitive and localizing operator algebras. Also I

investigate hyperinvariant subspaces.

Keyword(s): Transitive algebra, localizing algebra, hyperinvariant subspace.

2010 AMS Classification: 47L10, 47L45, 47A15

References:

1. Y. A. Abromovich, C.D. Aliprantis, An Invitation To Operator Theory, American

Mathematical Society, Rhode Island (2002).

2. C. D. Aliprantis, K. C. Border, Infinite Dimensional Analysis, A Hitchiker’s Guide, Second

Edition, Springer-Verlag, Berlin (1999).

3. J. B. Conway, A Course In Functional Analysis, Second Edition, Springer-Verlag, New

York, (1990).

4. V. I. Lomonosov, H. Radjavi, V. G. Troitsky, Sesquitransitive and Localizing Operator

Algebras, Integral Equations and Operator Theory, 60 (2008), 405-418.

5. B. P. Rynne, M. A. Youngson, Linear Functional Analysis, Springer-Verlag, London,

(2008).

6. V. G. Troitsky, Minimal vectors in arbitrary Banach spaces, Proc. American Mathematical

Society, 132(2004), 1177-1180.

109



1Anadolu University, Science Faculty, Department of Physics, Eskisehir,Turkey 2Dumlupınar University, Faculty of Art and Science, Department of Physics, Kütahya, Turkey


Reformulation of Compressible Fluid Equations in Terms of Biquaternions

Süleyman Demir1, Murat Tanışlı

1, Mustafa Emre Kansu

2

In relevant literature, although Maxwell’s equations of electromagnetism have been

expressed in many mathematical forms, the same is not true for their analogous equations in

fluid mechanics. In this work, a reformulation is proposed based on biquaternions for the

Maxwell type equations of compressible fluids stimulating the biquaternionic generalization

of electric and magnetic fields in electromagnetism. After reviewing the analogy between the

structure of electrodynamics and fluid dynamics, the biquaternionic expressions of the fluid

Maxwell equations have been derived. Furthermore, the field and wave equations for fluids

have been presented in a compact and simple way.

Keywords: Biquaternion, fluid equations, Maxwell equations, field equations

2010 AMS Classification: 76A02, 76W05, 11R52

References:

1. Logan J.G, Hydrodynamic analog of the classical field equations, Phys. Fluids, 5, 868-

869, 1962.

2. Troshkin O.V, Perturbation waves in turbulent media, Comp. Maths. Math. Phys., 33,

1613-1628, 1993.

3. Marmanis H, Analogy between the Navier-Stokes equations and Maxwell's equations:

Application to turbulence, Phys. Fluids, 10, 1428-1437, 1998.

4. Kambe T, A new formulation of equations of compressible fluids by analogy with

Maxwell's equations, Fluid Dyn. Res. 42, 055502, 2010.

5. Scofield D.F., Huq P., Fluid dynamical Lorentz force law and Poynting theorem-

introduction, Fluid Dyn. Res., 46, 055513, 2014.

6. Scofield D.F., Huq P., Fluid dynamical Lorentz force law and Poynting theorem-

derivation and implications, Fluid Dyn. Res., 46, 055514, 2014.

7. Thompson R.J., Moeller T.M., A Maxwell formulation for the equations of a plasma,

Phys. Plasmas, 19, 010702, 2012.

8. Tanışlı M, Demir S., Şahin N., Octonic formulations of Maxwell type fluid equations, J.

Math. Phys., 56, 091701, 2015.

9. Demir S., Tanışlı M., Hyperbolic octonion formulation of the fluid Maxwell equations, J.

Korean Phys. Soc., 68, 616-623, 2016.

10. Demir S., Uymaz A., Tanışlı M., A new model for the reformulation of compressible

fluid equations, Chin. J. Phys., 55, 115-126, 2017.

11. Demir S., Tanışlı M., Spacetime algebra for the reformulation of fluid field equations, J.

Geo. Meth. Mod. Phys., 14, 1750075, 2017.

110

13th Algebraic Hyper structures and its Applications (AHA2017)



2Bahçeşehir University, Department of Mathematic Engineering, İstanbul, Turkey


On The Trace Formula for a Differential Operator of Second Order with Unbounded

Operator Coefficients

Erdal GÜL

1 and Duygu ÜÇÜNCÜ

2

We investigate the spectrum of a differential operator of second order with unbounded

operator coefficients and two terms and we calculate the trace of this operator.

Keywords: Hilbert Space, Self-adjoint operator, Kernel operator, Spectrum, Essential

spectrum, Resolvent.

2010 AMS Classification: 47A10, 34L20, 34L05

References:

1. Adıgözelov E., "About the trace of the difference of two Sturm-Liouville operator with

operator coefficient," Iz. AN AZ SSR, seriya _z-tekn. i mat. nauk5, 20-24 (1976)

2. Chalilova R. Z., "On regularization of the trace of the Sturm-Liouville operator equation,"

Funks. Analiz, teoriya funksiy i ikpril.-Maha_ckala3, 154-161 (1976).

3. Maksudov . F. G., Bayramogluand M., Adıguzelov E., "On regularized trace of Sturm-

Liouville operator on a finite interval with unbounded operator coefficient," Dokl. Akad.

Nauk SSSR 30, 169-173 (1984).

4. Adıgüzelov E., Avcı H. and Gül E., "The trace formula for Sturm-Liouville operator with

Operator coefficient," JMP (Journal of Mathematical Physics) 42, No:6, 2611-2624 (2001).

5. Albayrak I., Baykal O. and Gül E., "Formula for the highly regularized trace of Sturm-

Liouville operator with unbounded operator coefficients wich has singularity", Turkish

Journal of Mathematics, Vol.25, No:2, pp.307-322 (2001)

6. Kato T., Perturbation Theory for Linear Operators (Berlin-Heidelberg-New York-Verlag,

1980).

7. Lysternikand L. A., Sobolev V. I, Elements of functional analysis, English Trans.

(NewYork: FredrickUngar, 1955).

8. Cohbergand I. C., Krein M. G., Introduction to the Theory of LinearNon-self adjoint

Operators, Translation of Mathematical Monographs, Vol. 18 (AMS, Providence, RI, 1969).

9. Maksudov F.G., Bairamoglu M. and Adigezalov E., "On asymptotics of spectrumand trace

of high order differential operator with operator coefficients, Doğa-TurkishJournal of

MathematicsVol 17, No: 2 , 113-128 (1993).

111





1Yildiz Technical University, Department of Mathematics, Esenler, İstanbul 34220, Turkey 2Gebze Technical University, Department of Mathematics, Gebze, Kocaeli 41400, Turkey


Practical Stability Analyses of Nonlinear Fuzzy Dynamic Systems of Unperturbed

Systems with Initial Time Difference

Mustafa Bayram Gücen1, Coşkun Yakar

2

In this work, we have investigated the practical stability of fuzzy differential systems

of unperturbed systems and we have established a comparison result. Some practical stability

theorems is presented; in the last section, we have a comparison result in practical stability of

fuzzy differential systems of unperturbed systems via a scalar differential equation.

Keyword(s): Initial time difference, practical stability, fuzzy differential equations

2010 AMS Classification: 34D10, 34D99

Reference(s):

1. Lakshmikantham, V. and Leela, S, Differential and Integral Inequalities, Vol. I.

Academic Press, New York, 1969.

2. Lakshmikantham, V. and Leela, S, Fuzzy differential systems and the new concept of

stability. Nonlinear Dynamics and Systems Theory, 1 (2) 111-119, 2001

3. Lakshmikantham, V. and Mohapatra, R. N. Theory of Fuzzy Differential Equations.

Taylor and Francis Inc. New York, 2003.

4. Yakar, C. and Shaw, M. D. , A comparison result and Lyapunov stability criteria with

initial time difference. Dynamics of Continuous, Discrete & Impulsive Systems. Series

A, vol. 12, no. 6, 731—737, 2005.

5. Li, A., Feng, E. and Li, S., Stability and boundedness criteria for nonlinear differential

systems relative to initial time difference and applications. Nonlinear Analysis: Real

World Applications 10, 1073—1080, 2009.

112





2 Iskenderun Technical University, Department of Computer Engineering, Hatay, Turkey


A Numerical Scheme for Solving Nonlinear Fractional Differential Equations in The

Conformable - Derivative

Sebahat Ebru DAŞ1, Sertan ALKAN

2

Fractional Calculus is a field that involves noninteger order differential and integral

operators. The history of fractional calculus dates back to the end of the 17th century. In 1695,

half-order derivative was mentioned in a letter from L’Hopital to Leibniz [1]. Since then,

many mathematicians have contributed to the development of fractional calculus. Therefore,

many definitions for the fractional derivative are available. The most popular definitions are

Riemann-Liouville and Caputo. Additionally, recently Khalil et at. [2] introduced a new

definition of fractional derivative called the Conformable Fractional Derivative.

In our work, Sinc-Collocation Method is presented to obtain the approximate solution

of the fractional order boundary value problem with variable coefficients in the following

form

𝜇2(𝑥)𝑦′′(𝑥) + 𝜇𝛼(𝑥)𝑦(𝛼)(𝑥) + 𝜇1(𝑥)𝑦′(𝑥) + 𝜇𝛽(𝑥)𝑦(𝛽)(𝑥) + 𝜇0(𝑥)𝑦(𝑥) + 𝑛(𝑥)𝑦𝑚(𝑥) = 𝑓(𝑥)

with boundary conditions

𝑦(𝑎) = 0 , 𝑦(𝑏) = 0

where 𝑦(𝛼) and 𝑦(𝛽) are the conformable fractional derivative for 1 < 𝛼 ≤ 2 and 0 < 𝛽 ≤ 1.

Keywords: Nonlinear differential equations, conformable –derivative


Reference(s):

1. Samko S.G., Kilbas A.A. , Marichev O.I., Fractional Integrals and Derivatives, Gordon

and Beach, Yverdon, 1993

2. Miller K., Ross B., An Introduction to The Fractional Calculus and Fractional Differential

Equations, New York: Wiley, 1993.

113





On Locally Convex Solid Riesz Spaces

Fatma ÖZTÜRK ÇELİKER1 and Pınar ALBAYRAK

1

An ordered vector space E is called a Riesz space if every pair of vectors has a

supremum and an infimum. A locally convex (solid) topology on a vector space is a linear

topology that has a base at zero consisting of convex (solid) sets. A subset S of Riesz space E

is said to be solid if |𝑢| ≤ |𝑣| and 𝑣 ∈ 𝑆 imply 𝑢 ∈ 𝑆. A linear topology 𝜏 on a Riesz space E

that is at the same time locally solid and locally convex will be called a locally convex-solid

topology. A locally convex-solid Riesz space (𝐸, 𝜏) is a Riesz space E equipped with a locally

convex-solid topology 𝜏.

On this study we consider invariant ideals on locally convex solid Riesz spaces for

positive operators.

Keyword(s): Invariant ideals, locally convex solid Riesz Spaces


References:

1. Abromovich Y.A., Aliprantis C. D., Burkinshaw O., An Invation to Operator Theory,

American Mathematical Society, 2002.

2. Abromovich Y.A., Aliprantis C. D., Burkinshaw O., The Invariant Subspace Problem:

Some Recent Advances, Rend. Istit. Mat. Univ. Trieste 29, 3-79, 1998.

3. Aliprantis C. D., Burkinshaw O., Positive Operators, Academic Press, 1985.

4. Aliprantis C. D., Burkinshaw O., Locally Solid Riesz Spaces with Applications to

Economics, American Mathematical Society, 2003.

114






On Weakly Compact-Friendly Operators

Pınar Albayrak1 and Fatma Öztürk Çeliker

1

A positive operator B:E E is said to be weakly compact-friendly if there exists a

positive operator in the commutant of B dominates a non-zero operator which in turn is

dominated by positive weakly compact operator. That is, B is weakly compact-friendly if and

only if there exists three non-zero operators R, C, K : E E with R, K positive and K

weakly compact such that

RB BR , Cx R x , and Cx K x

for each x E .

On this talk we generalized some well-known results of weakly compact-friendly

operators on Banach lattices.

Keyword(s): Invariant ideals, invariant subspaces, weakly compact-friendly operators

2010 AMS Classification: 47B65, 47A15

References:

1. Aliprantis, C.D. ,Burkinshaw, O., Positive Compact Operators on Banach Lattices, Math.

Z., 174, 289-298, 1980.

2. Aliprantis, C.D. ve Burkinshaw, O., Positive operators, Academic Pres, New

York/London,1985.

3. Gök, Ö. , Albayrak, P., On Invariant Subspaces of Weakly Compact-Friendly Operators,

International Journal of Contemporary Mathematical Sciences, Vol. 4, no:6, 259-266,2009.

115





Hirota Type Discretization Of Clebsch Equations

Murat Turhan1

The equations of motion of a rigid body in an ideal fluid is given by the following system:

{

�̇� = 𝐱 × 𝜕𝐻

𝜕𝒑

�̇� = 𝐱 × 𝜕𝐻

𝜕𝒙+ 𝒑 ×

𝜕𝐻

𝜕𝒑

where 𝐻 ∈ 𝐶∞(ℝ6, ℝ) is a quadratic polynomial in 𝐱 and 𝒑.

Applying bilinear method and using the gauge invariance and the time reversibility of the

equations, we get gauge-invariant bilinear difference equations. Finally, we derive the explicit

discrete system by considering Hirota bilinear transformation method and present sufficient

number of the discrete conserved quantities for integrability.

Keywords: Clebsch system, discretization, bilinear form, Gröbner basis

2010 AMS Classification: 70H99

References:

1. Abraham, R., Marsden, J.E. and Ratiu, T.S., Manifolds, Tensor Analysis, and

Applications, V.75 of Applied Mathematical Sciences, Springer-Verlag. 1988.

2. http://www.asir.org

3. Lesser, M., The Analysis of Complex Nonlinear Mechanical Systems: A Computer

Algebra Assisted Approach, World Scientific, Series A, Vol.17, 1995.

4. Hirota, R., Kimura, K. and Yahagi, H., How to find the conserved quantities of

nonlinear discrete equations, J.Phys.A:Math.Gen. 34, 10377–10386, 2001.

5. Zhivkov, A., Christov, O., Effective solutions of Clebsch and C. Neumann systems,

Sitzungsberichte der Berliner Mathematischen Gesellschaft, 217–242, 2001.

6. Perelomov, A.M., A few remarks about integrability of the equations of motion of a

rigid body in ideal fluid, Phys.Lett.A 80, no:2-3, 156–158, 1980.

116




2Bitlis Eren University, Department of Mathematics, Bitlis, Turkey

E-mail: [email protected] , [email protected], [email protected]

An h-deformation of the Superspace R(1|2) via a contraction

Salih Çelik 1, Sultan A. Çelik 1 and Fatma Bulut 2

A deformation of classical matrix (super) groups can be made using some facts known

from the classical. One way to obtain a deformation of classical (super) groups is to make a

deformation of (super) spaces first [1].

The one-parametric h-deformation of the algebra of coordinate functions on the

superspace 1| 2R via a contraction of the quantum superspace 1| 2qR is presented [2]. An

interesting case is that the deformation parameter h is Grassmann number.

It is well known that a matrix T in the supergroup 1| 2GL defines the linear

transformation : 1| 2 1| 2h hT R R . As a result of this we have R 1| 2hT X X .

So, the elements of the matrix T fulfill some relations. The bi-algebra structure of 1| 2hGL

is discussed.

Keywords: Quantum superspace, q-deformation, h-deformation, quantum supergroup, Hopf

superalgebra.

2010 AMS Classification: 17B37, 81R60

References:

1- Manin, Yu I., Multiparametric quantum deformation of the general linear supergroups,

Commun. Math. Phys. 123 (1989), 163-175.

2- Madore, J., An Introduction to Noncommutative Geometry and its Physical Applications,

Cambridge U. P., Cambridge, (1995).

3- Zakrzewski, S., A Hopf _-algebra of polynomials on the quantum SL(2,R) for a unitary R-

matrix, Lett. Math. Phys. 22 (1991), 287-289.

4- Ohn, C., A _-product of SL(2) and the corresponding nonstandard quantum Uh(sl(2)), Lett.

Math. Phys. 25 (1992), 85-88.

5- Kupershmidt, B.A., The quantum group GLh(2), J. Phys. A 25 (1992), L1239-1244.

6- Aghamohammadi, A., M. Khorrami and A. Shariati, h-deformation as a contraction of q-

deformation, J. Phys. A: Math. Gen. 28 (1995), L225-L231.

117







2Bitlis Eren University, Department of Mathematics, Bitlis, Turkey

E-mail: [email protected] , [email protected], [email protected]

7- Celik, S., Bicovariant differential calculus on the superspace Rq(1j2), J. Alg. and Its Appl.

15, No.9 (2016), 1650172 (17 pages)

8- Aizawa, N. and R. Chakrabarti, Noncommutative geometry of super-jordanian OSph(2/1)

covariant quantum space, J. Math. Phys. 45 (2004), 1623-1638.

9- Kac, V.: Lie Superalgebras, Adv. in Math. 26 (1977), 8-96.

10- Celik, S. and Bulut, F., A differential calculus on superspace Rh(1j2) and related topics,

Adv. Appl. Clifford Algebras 27 (2017), 1019-1030.

Salih C¸ elik, Sultan C¸ elik aa B

118




13th Algebraic Hyperstructures and its Applications (AHA2017) 24-27 July, 2017, Istanbul-Turkey

Euler-Savary’s Formula on Dual Plane

Mücahit Akbıyık1, Salim Yüce2

In this work, one three Dual planes, of which two of them are moving and the other

one is fixed, are considered during the one parameter motion. In each motion; the velocities,

the relation between the velocities and the rotation poles were calculated. In addition, Euler-

Savary formula, which gives the relationship between the curvature of pole curves and

trajectory curve, were given by two different methods.

Keywords: Dual Plane, Euler Savary’s Formula, Kinematics.

2010 AMS Classification: 53A17,53A35, 53A40.

References:

1. Edmund Taylor Whittaker, A Treatise on the Analytical Dynamics of Particles and RigidBodies, Cambridge University Press. Chapter 1, 1904.

2. Joseph Stiles Beggs, Kinematics, Taylor and Francis. p. 1, 1983.3. Thomas Wallace Wright, Elements of Mechanics Including Kinematics, Kinetics and

Statics. E and FN Spon. Chapter 1, 1896. 4. A. Biewener, Animal Locomotion, Oxford University Press, 2003.5. Yaglom, I.M., A simple non-Euclidean Geometry and its Physical Basis, Springer-Verlag,

New York, 1979. 6. Yaglom I.M., Complex Numbers in Geometry, Academic Press, New York, 1968.7. Alexander J.C., Maddocks J.H., On the Maneuvering of Vehicles, SIAM J. Appl. Math.

48(1): 38-52,1988. 8. Buckley R., Whitfield E.V., The Euler-Savary Formula, The Mathematical Gazette

33(306): 297-299, 1949. 9. Dooner D.B., Griffis M.W., On the Spatial Euler-Savary Equations for Envelopes, J. Mech.

Design 129(8): 865-875, 2007. 10. Ito N., Takahashi K.,Extension of the Euler-Savary Equation to Hypoid Gears, JSME

Int.Journal. Ser C. Mech Systems 42(1): 218-224, 1999. 11. Pennock G.R., Raje N.N.,Curvature Theory for the Double Flier Eight-Bar Linkage,

Mech. Theory 39: 665-679, 2004. 12. Blaschke, W., and Müller, H.R., Ebene Kinematik, Verlag Oldenbourg, München, 1959.13. A. A. Ergin, On the one-parameter Lorentzian motion,Comm. Fac. Sci. Univ. Ankara,

Series A 40, 59--66, 1991.14. A.A.Ergin, Three Lorentzian planes moving with respect to one another and pole points,

Comm. Fac. Sci.Univ. Ankara, Series A 41,79-84,1992.1Yildiz Technical University,Department of Mathematics,Istanbul, Turkey 2Yildiz Technical University,Department of Mathematics,Istanbul, Turkey E-mails:[email protected], [email protected]

119

13th Algebraic Hyperstructures and its Applications (AHA2017) 24-27 July, 2017, Istanbul-Turkey

15. I. Aytun, Euler-Savary formula for one-parameter Lorentzian plane motion and itsLorentzian geometrical interpretation, M.Sc. Thesis, Celal Bayar University, 2002.

16. T. Ikawa, Euler-Savary's formula on Minkowski geometry, Balkan Journal of Geometryand Its Applications, 8 (2), 31-36, 2003.

17. Otto Röschel, Zur Kinematik der isotropen Ebene., Journal of Geometry, 21, 146--156,1983.

18. Akar, M., Yüce, S. and Kuruoglu, N, One parameter planar motion in the Galilean plane,International Electronic Journal of Geometry, 6(1), 79-88, 2013.

19. Akbiyik, M., Yüce, S., The Moving Coordinate System And Euler-Savary's Formula ForThe One Parameter Motions On Galilean (Isotropic) Plane, International Journal ofMathematical Combinatorics(2), 88-105.

20. Kuruoğlu, N., Tutar, A., Düldül, M., On the moving coordinate system on the complexplane and pole points, Bulletin of Pure and Applied Sciences, Vol. 20 E,No1, 1-6, 2001.

21. Masal, M., Tosun, M., Pirdal, A. Z., Euler Savary formula for the one parameter motionsin the complex plane

1Yildiz Technical University,Department of Mathematics,Istanbul, Turkey 2Yildiz Technical University,Department of Mathematics,Istanbul, Turkey E-mails:[email protected], [email protected]

120




E-mail: [email protected] , [email protected]

Mistakes and Misconceptions Regarding to Natural Numbers on Secondary

Mathematics Education

Ayten Özkan1 and Erdoğan Mehmet Özkan

1

Mistakes and misconceptions regarding to natural numbers of 12th class students and

whether these mistakes and misconceptions demonstrated any significant difference

depending on the gender has been investigated in this study. This study was carried out with

60 students at 12th class who have being educated in Anatolian High Schools located in

Istanbul City Fatih Province in the education training year of 2017-2018 after completing of

the natural number subject. Cronbach Alpha coefficient reliability of the Diagnosis Test was

found as 0,90. An expert opinion was obtained for the validity. The SPSS 15 pack program

was used in order to solve the data obtained by Diagnosis Test which composed by open-

ended questions. Qualitative and quantitative researching methods were utilized in this study.

The answers given by the students were examined individually and the answers of the

students were evaluated in categories such as “correct”, “mistake”, “empty” and

“misconception”, then distribution of these students’ answers into percentage and frequency

categories were determined. Also samples within all determined mistakes and misconceptions

transferred into the computer via scanner and were submitted in findings. At the end of the

investigation, it has been determined that students had a lot of mistake and misconceptions

regarding to natural number, features belong to exponentiation, base arithmetic, prime

numbers, relative prime numbers, prime factorization of a natural number, positive dividends

of a natural number and factorial. Also these mistakes and misconceptions were determined as

not demonstrating a significant difference depending on the genders.

Keywords: Mathematics teaching, misconception, common mistakes, natural numbers

2010 AMS Classification: 97C10, 97C70

References:

1-Ercan, B. “Evaluation of the Information Related to the Concept of the Integers of the Seventh

Grade Primary School Students”, Master Thesis, Çukurova University, Social Sciences Institute,

(2010).

2-Güner, N. and Alkan, V. “Errors in Primary and Secondary Students' Answering 2010 YGS

Mathematics Questions”, Pamukkale University Journal of Education, 2, 30, (2011) ,125-140.

3- Ubuz, B., “Problems and Misconceptions of 10th and 11th Grade Students on Basic Geometry

Issues”, Hacettepe University Journal of Education, 16, 17, (1999) 95 –104.

4- Zembat İ.Ö. “What is Misconception? ", Misconceptions of Mathematical Concept and Solution

Proposals”, Edit: Özmantar,M.F. , Bingölbali, E.and Akkoç, H.,Pegem Academy:Ankara, (2010).

121






2Yildiz Technical University, Marine Machinery, Istanbul,Turkey

E-mails:[email protected], vardar@ yildiz.edu.tr

On a Fuzzy Application of the Particulate Matter Estimation

Filiz Kanbay1, Nurten Vardar

2

In this study, Particulate matter PM from transit vessels passing through the

Bosphorus which connects Black sea and sea of Marmara with the length 12 sea knot are

calculated by using fuzzy inference system in MATLAB. Total particulate matters from ships

are expressed surfaces and these results allow the analysis of the data gross tone and the type

of ships.

Keywords: Particulate matter, fuzzy, surface, ship

2010 AMS Classification: 65D18, 68T27, 93B99

References:

1. Deniz C., Yalçın D. “Estimating Shipping Emissions in the Region of the Sea of Marmara,

Turkey” Science of the Total Environment 390(2008)255-261

2. Trozzi, C., Vaccoro,R., 1998. Methodologies for Estimating Air Pollutant Emissions From

Ships. Techne Report MEET. (Methodologies for Estimating Air Pollutant Emissions from

Transport)RF98

3. Kesgin, U., Vardar, N., 2001. A study on Exhaust gas Emissions from Ships in Turkish

Straits Atmospheric Environment 35, pp. 1863-1870.

4. Kılıç, A., 2009. Marmara Denizi’nde Gemilerden Kaynaklanan Egzoz Emisyonları, BAÜ

FBE Dergisis Cilt:11,Sayı:2, pp.124-134

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