International Symposium on Aggregation andStructures

Book of Abstracts

ISAS 2016

ISAS

Editors: Gergely Kiss, Jean-Luc Marichal, and Bruno Teheux

ISBN 978-99959-0-241-4

Preface

Over the last decades the theory and applications of aggregationfunctions have become more and more important in a large numberof disciplines such as applied mathematics, computer sciences, statis-tics, economics, and engineering sciences. The considerable amountof knowledge that has been collected up to now in this area has ledthe researchers to extend their investigation to wider fields, includingaggregation of more abstract structures such as ordered sets, struc-tured data, etc. The ‘International Symposium on Aggregation onBounded Lattices’ (ABLAT 2014, Trabzon, Turkey) was a first con-ference organized in this direction and provided a medium for theexchange of ideas between theoreticians and practitioners in aggre-gation on lattices and related areas.

The ‘International Symposium on Aggregation and Structures’(ISAS 2016, Luxembourg, Luxembourg) aims at pursuing this tra-dition of investigating the aggregation problem in a very wide sense.More specifically this symposium constitutes a forum for presentingthe latest trends in both aggregation on structures and aggregationof structures. ISAS 2016 is organized by the Mathematics ResearchUnit of the University of Luxembourg with the support of the ILIASlaboratory. We hope that this initiative will be followed in the futureand that more and more researchers will join this community.

We are very grateful to all those who contributed to the successof ISAS 2016. In particular our special thanks go to the authors andreviewers of the submitted abstracts. We have selected 28 submis-sions that fit best to the objectives of this symposium. Also, fourinvited contributions provide new trends in extended aggregationtheory. In order to help the attendees and participants interact withthe various presentations we have not organized any parallel session.

We wish to all participants a very pleasant symposium full offruitful projects of collaborations.

Gergely Kiss (co-chair)Jean-Luc Marichal (co-chair)

Bruno Teheux (chair)

Scientific Committee

Bernard De BaetsMiguel CouceiroRadko Mesiar

Program Committee

Jamal AtifMichal BaczynskiGleb BeliakovHumberto BustinceDenis BouyssouMiguel CouceiroBernard de BaetsJózsef DombiDidier DuboisJozo DujmovicPawel DrygasJános FodorMarek GagolewskiMichel GrabischRadomir HalašFrancisco HerreraEsteban IndurainMartin KalinaFunda KaraçalGergely KissErich Peter Klement

Anna KolesarovaCarlos Lopez-MolinaGyula MaksaRicardo Alberto M. PereiraJean-Luc MarichalRadko MesiarGaspar MayorJavier MonteroMirko NavaraSergei OvchinnikovEndre PapGabriella PigozziMarc PirlotHenri PradeSusanne Saminger-PlatzAndrea StupnanovaBruno TeheuxVicenc TorraLeon ven der TorreEmil Weydert

Organizing Committee

Gergely KissJean-Luc MarichalBruno Teheux (chair)

Contents

Preface 3Invited SpeakersMarek GagolewskiPenalty-Based Aggregation of Complex Data and Their Applica-tions in Data Analysis 8Michel GrabischInteger means and nonassociative calculus 9Carlos Lopez-MolinaBeyond pixels: when visual information escapes the matrix 10Gabriella PigozziCompatible group decisions 10Contributed TalksEmel AşıcıAn Order Obtained from Nullnorms and Its Properties 11Michał Baczyński, and Katarzyna MiśDirectional monotonicity of fuzzy implications 12Slavka Bodjanova, and Martin KalinaSome constructions of uninorms and nullnorms on bounded lat-tices 14Quetin Brabant, Miguel Couceiro, Fabien Labernia, and AmedeoNapoliA Dimensionality Reduction Approach for Qualitative PreferenceAggregation 16Humberto Bustince, Javier Fernandez, Radko Mesiar, AnnaKolesárová, Benjamen Dedregal, Graçaliz P. Dimuro, and EdurneBarrenecheaDirectional monotonicity and ordered directionalmonotonicity 18Juan C. CandealA Generalized Form of a Functional Equation Related to Scien-tific Laws 20Giulio Caperna and Giovanna BoccuzzoThe HOGS procedure: a sampling approach to aggregate subgroupsof a poset 21

Marta CardinPreferences and aggregation operators over property spaces 23Gül Deniz Çaylı, Funda Karaçal, and Radko MesiarOn a new class of uninorms on bounded lattices 25Jacek ChudziakQuasideviation means in insurance 27Kevin Dalleau, Miguel Couceiro, Marie-Dominique Devignes,Chedy Raisi, and Malika Smail-Tabonne,Using aggregation functions on structured data: a use case in theFIGHT-HF project 2!Bernard De Baets, and Raúl Pérez-FernándezRanking rules characterized by means of monometrics and con-sensus states 30József Dombi, and O. Csiszár,Self De Morgan identity in nilpotent systems and a unified de-scription of nilpotent operators 33Paweł DrygaśProperties of uninorms and its generalization 36Soma Dutta, Mihir K. Chakraborty, and Andrzej SkowronAggregation in logical consequence: revisiting from gradedcontext 37José Luis García-Lapresta, and David Pérez-RománA Group Decision Making Procedure in the Context ofNon-Uniform Qualitative Scales 39Debashree Guha, and Radko MesiarInterval-valued Extended Bonferroni Mean Operator and Its Ap-plications in Multi-criteria Decision Making 41Radomir Halaš, Jan Kühr, and Jozef PócsLattices with a smallest set of aggregation functions 43Radomir Halaš, Radko Mesiar, and Jozef PócsGeneralized comonotonicity and a characterization of discreteSugeno integrals on bounded distributive lattices 44Mehmet Akif Ince, Funda Karaçal, and Radko MesiarSome Notes on Nullnorms on Bounded Lattices 46Simon JamesElicitation of fuzzy partial orders from incompletepreferences 48

Balasubramaniam JayaramA New Ordering based on Fuzzy Implications 49Funda Karaçal, Ümit Ertugrul, and Gül Deniz ÇaylıOrdinal sum of t-norms on bounded lattices 51Juan KawabeA perturbative approach to convergence theorems for nonlinearintegrals 53Anna Kolesárová, and Radko MesiarAggregation on some posets of aggregation functions 55Anna KrólSome properties of (C, I)-equivalences 57Radko Mesiar, and Andrea StupňanováExtremal weighted aggregation 58Raúl Pérez-Fernández, and Bernard De BaetsMonotonicity: a corner stone of social choice theory 60

Penalty-Based Aggregation of Complex Data andTheir Applications in Data Analysis

Marek Gagolewski1,2

1 Systems Research Institute, Polish Academy of Sciences,ul. Newelska 6, 01-447 Warsaw, Poland

2 Faculty of Mathematics and Information Science, Warsaw University of Technology,ul. Koszykowa 75, 00-662 Warsaw, Poland

gagolews@ibspan.waw.pl, http://www.gagolewski.com

Abstract

Since the 1980s, studies of aggregation functions most often focus on the con-struction and formal analysis of diverse ways to summarize numerical lists withelements in some real interval. Quite recently, we also observe an increasinginterest in aggregation of and aggregation on generic partially ordered sets.

However, in many practical applications, we have no natural ordering of givendata items. Thus, in this talk we review various aggregation methods in spacesequipped merely with a semimetric (distance, see [3]). These include the conceptof such penalty minimizers as the centroid, 1-median, 1-center, medoid, and theirgeneralizations – all leading to idempotent fusion functions (see, e.g., [1]). Specialemphasis is placed on procedures to summarize vectors in Rd for d ≥ 2 (e.g.,rows in numeric data frames) as well as character strings (e.g., DNA sequences),but of course the list of other interesting domains could go on forever (rankings,graphs, images, time series, and so on).

We discuss some of their formal properties, exact or approximate (if theunderlying optimization task is hard) algorithms to compute them (see, e.g., [4])and their applications in clustering and classification tasks [2,5,6].

References

1. Beliakov, G., Bustince, H., Calvo, T.: A Practical Guide to Averaging Functions.Springer (2016)

2. Cena, A., Gagolewski, M.: Fuzzy k-minpen clustering and k-nearest-minpen classi-fication procedures incorporating generic distance-based penalty minimizers. Com-munications in Computer and Information Science 611, 445–456. Springer (2016)

3. Deza, M.M., Deza, E.: Encyclopedia of Distances. Springer (2014)4. Gagolewski, M.: Data Fusion: Theory, Methods, and Applications. Institute of Com-

puter Science, Polish Academy of Sciences, Warsaw, Poland (2015)5. Gagolewski, M., Bartoszuk, M., Cena, A.: Genie: A new, fast, and outlier-resistant

hierarchical clustering algorithm. Information Sciences 363, 8–23 (2016)6. Gagolewski, M., Cena, A., Bartoszuk, M.: Hierarchical clustering via penalty-based

aggregation and the Genie approach. In: Proc. MDAI’16, Lecture Notes in ArtificialIntelligence, Springer (2016), in press

8

Integer means and nonassociative calculus

Michel Grabisch(with Miguel Couceiro)

Paris School of Economics, Université Paris I Panthéon-Sorbonne,Centre d’économie de la Sorbonne, 106 – 112, bd de l’Hôpital, 75013 Paris,France

michel.grabisch@univ-paris1.fr

Abstract

Integer-valued means aggregate values in Z, and satisfy symmetry (anonymity),internality, monotonicity and decomposability in the sense of Kolmogoroff. It hasbeen recently proved by Bennett et al. that all integer means are extremal, i.e.,the result is a function of the minimum and maximum entries. This result showsthat the restriction to integers prevents from using all entries, and can be seen asa negative result. We try in this work to overcome this limitation, by weakeningthe decomposability axiom in order to use not only extremal values but alsosecond extremal values, etc. Besides, the extension of the maximum on Z leadsto the symmetric maximum, which is a nonassociative operator. By defining rulesof computation (i.e., systematic ways of putting parentheses), one can overcomenonassociativity. It turns out that these extended symmetric maximum operatorsprecisely use extremal elements up to some rank and can be considered as anew family of integer means. We identify which type of weak decomposabilityis satisfied by the symmetric maximum operators, and give axiomatizations ofthese operators.

9

Beyond pixels: when visual information escapesthe matrix

Carlos Lopez-Molina

GIARA, Universidad Pública de NavarraPamplona 31006, Spain carlos.lopez@unavarra.es

Abstract

Digital imagery has always be represented as a matrix of pixels, each of themholding a tone. Although innovation has led to a wide variety of tasks, goals andapplications, very few authors have analyzed the representation and processingof images in a shape other than matrices. Although mathematically convenient,we find this representation inadequate to mimic human behaviour. In this talkwe analyze the superpixel paradigm, from its biological roots to its requirementsin terms of mathematical modelling. We pay special attention to the relationshipwith soft computing and non-standard data fusion.

Compatible group decisions

Gabriella Pigozzi(with E. Awad, M. Caminada, M. Podlaszewski, and I. Rahwan)

LAMSADE - Université Paris DauphinePlace du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France

gabriella.pigozzi@dauphine.fr

An inconsistent knowledge base can be abstracted as a set of arguments anda defeat relation among them. There can be more than one consistent way toevaluate such an argumentation graph. Collective argument evaluation is theproblem of aggregating the opinions of multiple agents on how a given set ofarguments should be evaluated. It is crucial not only to ensure that the out-come is logically consistent, but also satisfies measures of social optimality andimmunity to strategic manipulation. This is because agents have their individ-ual preferences about what the outcome ought to be. In this talk I will presentthree argument-based aggregation operators and they analysis with respect toPareto optimality and strategy proofness under different general classes of agentpreferences.

10

An Order Obtained fromNullnorms and Its Properties

Emel Asıcı

Department of Software Engineering, Faculty of Technology, Karadeniz TechnicalUniversity, 61830 Trabzon, TurkeyE-mail: emelkalin@hotmail.com

Abstract. In this paper, an order induced by nullnorms on boundedlattices is given and discussed. We define the set of incomparable ele-ments with respect to the order induced by a nullnorm. Also, by definingsuch an order, an equivalence relation on the class of nullnorms is definedand this equivalence is deeply investigated.

References

1. Asıcı, E., Karacal, F.: On the T -partial order and properties. Inf.Sci. 267, 323-333 (2014)

2. Asıcı, E., Karacal, F.: Incomparability with respect to the triangular order.Kybernetika. 52, 15-27 (2016)

3. Birkhoff, G.: Lattice Theory. 3 rd edition. Providence. 1967.4. Calvo, T., De Baets, B., Fodor, J.: The functional equations of Frank and

Alsina for uninorms and nullnorms. Fuzzy Sets Syst. 120, 385-394 (2001)5. Drewniak, J., Drygas, P., Rak, E.: Distributivity between uninorms and null-

norms. Fuzzy Sets Syst. 159, 1646-1657 (2008)6. Drygas, P.: A characterization of idempotent nullnorms. Fuzzy Sets Syst. 145,

455-461 (2004)7. Grabisch, M., Marichal, J.-L., Mesiar, R., Pap, E.: Aggregation Functions.

Cambridge University Press. 20098. Karacal, F., Asıcı, E.: Some notes on T -partial order. J. Inequal. Appl. 2013,

219 (2013)9. Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Pub-

lishers. Dordrecht 200010. Mas, M., Mayor, G., Torrens, J.: t-operators. Int. J. Uncertain. Fuzz. Knowl.-

Based Syst. 7, 31-50 (1999)11. Mitsch, H.: A natural partial order for semigroups. Proceedings of the American

Mathematical Society. 97, 384-388 (1986)

11

Directional monotonicity of fuzzy implications

Micha l Baczynski and Katarzyna Mis

Institute of Mathematics, University of Silesia,40-007 Katowice, ul. Bankowa 14, Poland

michal.baczynski@us.edu.pl

kmis@us.edu.pl

Standard monotonicity is one of the key properties of aggregation opera-tions. For example t-norms, t-conorms and copulas are increasingly monotonein each variable (cf. [3]). But fuzzy implication functions, which are very usefulin fuzzy logic and fuzzy control, are hybrid monotonic – they are decreasing inthe first variable and increasing in the second one. Our motivation for this workis the article [2], where the authors discussed directional monotonicity for fusionfunctions.

Definition 0.1 ([2]). Let F : [0, 1]n → [0, 1] and Rn 3 r = (r1, . . . , rn) 6=(0, . . . , 0).

(i) F is r-increasing, if for all x ∈ [0, 1]n and all c > 0 such that x+ cr ∈ [0, 1],it holds that

F (x + cr) ≥ F (x).

(ii) F is r-decreasing, if for all x ∈ [0, 1]n and all c > 0 such that x+ cr ∈ [0, 1],it holds that

F (x + cr) ≤ F (x).

Fuzzy implications (see [1]) are in fact (1, 0)-decreasing and (0, 1)-increasingfunctions. Please note that the (ε, ε)-increasing fuzzy implications, for ε > 0,have been investigated in [4]. This type of increasingness is also called weakincreasingness (see [5]).

Definition 0.2. A fuzzy implication I is said to be special, if for any ε > 0and for all x, y ∈ [0, 1] such that x + ε, y + ε ∈ [0, 1] the following condition issatisfied:

I(x, y) ≤ I(x + ε, y + ε).

In [4] the authors have obtained a characterization of general binary opera-tions whose residuals become special (in particular R-implications). In our con-tribution we present main results connected with special implications, discussdirectional monotonicity for fuzzy implications and we analyze (ε1, ε2)-increasingfuzzy implications and also (ε, ε)-decreasing fuzzy implication functions.

Key words: aggregation operations; functional equations; fuzzy implications;directional monotonicity

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2 Micha l Baczynski and Katarzyna Mis

References

1. Baczynski, M., Jayaram, B.: Fuzzy Implications, volume 231 of Studies in Fuzzinessand Soft Computing, Springer, Berlin Heidelberg (2008)

2. Bustince, H., Fernandez, J., Kolesarova, A., Mesiar, R.: Directional monotonicityof fusion functions. European J. Oper. Res. 224 (2015) 300–308

3. Grabisch, M., Marichal, J.-L., Mesiar, R., Pap, E.: Aggregation functions. Cam-bridge, Cambridge University Press (2009)

4. Jayaram, B., Mesiar, R.: On special fuzzy implications. Fuzzy Sets and Systems 160(2009) 2063–2085

5. Wilkin, T., Beliakov, G: Weakly monotone aggregation functions. Int. J. IntelligentSyst. 30 (2015) 144–165

13

Some constructions of uninorms and nullnorms

on bounded lattices

Slavka Bodjanova and Martin Kalina

1 Texas A&M University-Kingsville, Department of MathematicsMSC 172, Kingsville TX 78363, U.S.A.

kfsb000@tamuk.edu2 Slovak University of Technology in Bratislava

Faculty of Civil Engineering, Department of MathematicsRadlinského 11, Sk-810 05 Bratislava, Slovakia

kalina@math.sk

This contribution is a continuation of papers [1, 4, 5]. In [1] we have con-structed uninorms on some special types of bounded lattices. In [4] a pre-ordergenerated by uninorms was introduced. This pre-order was inspired by a partialorder induced by t-norms which was suggested by Karaçal and Kesicio§lu in[6]. Finally, as shown in [5], it is possible to construct operations on a latticeL which are both, proper uninorms and proper nullnorms, if there exist incom-parable elements b1 ∈ L and b2 ∈ L such that the following conditions areful�lled

b1 ∧ b2 = 0, b1 ∨ b2 = 1,

(∀x ∈ L)(x = (x ∧ b1) ∨ (x ∧ b2)).

We will discus other possible conditions under which it is possible to con-struct such operations. We will be interested also in (pre-)order induced by theseoperations.

In [2] the authors showed that on the unit interval there is no proper uninormsolving the well-known Frank functional equation (see [3]). We point out thatthis is not the case when considering uninorms on [0, 1]n for n > 1.

Acknowledgments. Martin Kalina has been supported from the Science andTechnology Assistance Agency under contract No. APVV-14-0013, and from theVEGA grant agency, grant number 2/0069/16.

References

1. Bodjanova, S., Kalina, M.: Construction of uninorms on bounded lattices. In: SISY2014, IEEE 12th International Symposium on Intelligent Systems and Informatics,Subotica, Serbia, 61�66 (2014).

2. Calvo, T., De Baets, B., Fodor, J.: The functional equations of Frank and Alsinafor uninorms and nullnorms. Fuzzy Sets and Systems 120, 385�394 (2001).

3. Frank, M.: On the simultaneous associativity of F (x; y) and x + y − F (x; y). Ae-quationes Mathematicae 19, 194�226 (1979).

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2 S. Bodjanova, M. Kalina

4. Hlin¥ná, D., Kalina, M., Krá©, P.: Non-representable uninorms. In: EUROFUSE2013, Uncertainty and Imprecision Modelling in Decision Making, 131�138. Serviciode Publicaciones de la Universidad de Oviedo, Oviedo (2013)

5. Kalina, M.: On uninorms and nullnorms on some bounded lattices. (submitted)6. Karaçal, F., Kesicio§lu, M. N.: A t-partial order obtained from t-norms. Kyber-

netika 47, No. 2, 300�314 (2011)

15

A Dimensionality Reduction Approach for QualitativePreference Aggregation

Quentin Brabant1, Miguel Couceiro1, Fabien Labernia2, Amedeo Napoli1

1 LORIA (CNRS, Inira Nancy Grand Est - Universite de Lorraine{quentin.brabant, miguel.couceiro, amedeo.napoli}@inria.fr

2 LAMSADE (CNRS - Universite Paris-Dauphine)fabien.labernia@dauphine.fr

1 Qualitative preference aggregation models

In this paper we briefly present a method for reducing the dimensionality of data in aqualitative preference aggregation framework. For a more complete description of thisapproach, see [4]. For an alternative approach based on rough sets theory, see [1].

We consider the following setting. X is a set of alternatives that are evaluated ac-cording to a set of criteria represented by there indices: [n] = {1, . . . , n}. For an alter-native x ∈ X we denote by (x1, . . . , xn) ∈ Ln the tuple of the evaluations of x in eachcrieterion. L is called the evaluation space, and is a distributive lattice for which wedenote respectively by 0 and 1 the minimal and maximal element. We consider a binarypreference relation 4 between the alternatives that can be expressed in terms of a utilityfunction:

∀x, y ∈ X : x 4 y ⇒ U(x) ≤ U(y),

where U : X → L associates a global evaluation on L to each alternative, and isobtained through the aggregation of the evaluations in criteria by a Sugeno integralSµ : Ln → L. In other words we have U(x) = Sµ(x1, . . . , xn). The Sugeno integraldefined over distributive lattices [3], is expressed

Sµ(x1, . . . , xn) =∨I⊆[n]

µ(I)∧i∈I

xi,

where µ : 2[n] → L a capacity, that is to say a non-decreasing set function on [n], withµ(∅) = 0 and µ([n]) = 1. Capacities (and Sugeno integrals) are defined by a value on Lfor each subset of [n], and therefore carry an intrinsic complexity, that grows exponen-tially with n. We now consider a set D = {(x1, y1), . . . , (xm, ym)} ⊆ Ln × L, whereeach xi = (xi1, . . . , x

in) ∈ Ln is a tuple of evaluations in n critieria, and yi is a utility

value associated to xj . We want to learn a Sugeno integral Sµ that generalizes thesedata. Ideally this function would be such that Sµ(xj) = yj for any j ∈ {1, . . . ,m}.However, it is very common that no such function exists: in that case D is said to beinconsistent, and we aim at learning a Sugeno integral that realizes the prediction of yj

for each element, with an error as low as possible. Because of the nature of capacities,this optimization problem is on 2n variables, and is therefore hard to solve when a highnumber of criteria is considered.

16

2 Dimensionnality reduction based on quality measure

By a quality measure over D we mean a degree with which D satisfies a certain hy-pothesis. In this presentation we consider two of such measures.

The first quality measure is the monotonicity degree, that is, the ratio of pairs {i, j} ⊆{1, . . . ,m} that satisfy the following condition

yi > yj ⇒ ∃k ∈ [n] : yik > yjk.

This condition can be seen as a generalization of the Pareto condition to partially or-dered evaluation spaces. The second quality measure is the compatibility degree, thatis, the ratio of pairs satisfying the the condition

∃Sµ : [Sµ(xi) = yi and Sµ(xj) = yj ]. (1)

This condition is justified by results from [2] that apply only when L is totally ordered.Indeed it can be shown that D is consistent if and only if (2) is true for any pair fromD . Moreover, for a given pair this condition can be checked in a linear time w.r.t. n.Hence, provided that L is totally ordered, the compatibility degree is both theoreticallymeaningful and practically interesting. If L is not totally ordered, the monotonicitydegree is the quality measure that makes sense.

The principle of the algorithm for dimensionality reduction that we propose is toiteratively remove a criterion, in order to minimize the decrease of the quality of thedataset at each step. Criteria are deleted until it is impossible to remove a criterionwithout decreasing the quality of the data below a certain ratio α. This algorithm wastested on empirical data 1 and allowed a reduction of the number of criteria from 7 to 3.Aggregation models trained on original data and on data reduced to 3 criteria showedto have similar accuracy. On the other hand, models trained on data with only 2 criterialeft had significantly worse accuracy, suggesting that a reduction to 3 criteria constitutesthe best compromise between simplicity and accuracy for these data.

Future research work should include further empirical studies and should aim todetermining a procedure for deciding the optimal value of α, currently being set byhand.

References1. Greco, S., Matarazzo, B., Slowinski, R.: Rough sets methodology for sorting problems in

presence of multiple attributes and criteria. European Journal of Operational Research, 138(2),247–259 (2002)

2. Prade, H., Rico, A., Serrurier, M., Raufaste, E.: Elicitating Sugeno integrals: Methodologyand a case study. Proceedings of the 10th European Conference Symbolic and QuantitativeApproaches to Reasoning with Uncertainty, 712–723 (2009)

3. Couceiro, M., Marichal, J.-L.: Characterizations of discrete Sugeno integrals as polynomialfunctions over distributive lattices. Fuzzy Sets and Systems 161(5), 694–707 (2010)

4. Brabant, Q., Couceiro, M., Labernia, F., Napoli, A.: Une approche de reduction de dimen-sionnalite pour l’agregation de preferences qualitatives. Proceedings of the 16th Conferencefrancophone sur l’Extraction et la Gestion des Connaissances, 345–350 (2016)

1Tripadvisor: http://sifaka.cs.uiuc.edu/˜wang296/Data/index.html

17

Directional monotonicity and ordered directionalmonotonicity

H. Bustincea,b?, J. Fernandeza,b, R.Mesiarc, A. Kolesarovad,B. Bedregale, G. P. Dimurof , and E. Barrenecheaa,b

a Dept. de Automatica y Computacion, Universidad Publica de Navarra,Campus Arrosadia, s/n, 31.006 Pamplona, Spain

b Institute of Smart Cities, Universidad Publica de Navarra,Campus Arrosadia, s/n, 31.006 Pamplona, Spain

c Slovak University of Technology,Radlinskeho 11, Bratislava, Slovakia

dInstitute of Information Engineering, Automation and Mathematics, SlovakUniversity of Technology,

Radlinskeho 9, Bratislava, SlovakiaeDepartamento de Informatica e Matematica Aplicada, Universidade Federal do Rio

Grande do Norte,Campus Universitario, s/n, Lagoa Nova, CEP 59078-970, Natal, Brazil

f Centro de Ciencias Computacionais, Universidade Federal do Rio Grande,Av. Italia, km 08, Campus Carreiros, 96201-900 Rio Grande, Brazil

bustince@unavarra.es

fcojavier.fernandez@unavarra.es

anna.kolesarova@stuba.sk

bedregal@dimap.ufrn.br

gracaliz@furg.br

Monotonicity is one of the key features of aggregation functions ([3]) both inthe real setting and when dealing with extensions of fuzzy sets. Its requirementis natural in many applied fields, as, for instance, image processing or decisionmaking. However, in some cases such requirement can be too strong. Consider,for instance, the case of the mode in image processing. Although it provides avery good tool to remove some kinds of noise, it is not monotone, so it can notbe considered to be an aggregation function ([1]). In this talk, and based on theidea of directional monotonicity [2], we introduce the notion of a pre-aggregationfunction([4]), as a function which satisfies the same boundary conditions as anaggregation function but for which only monotonicity along a fixed directionrather than monotonicity along any direction is required. In this sense, pre-aggregation functions encompass many relevant examples of functions which donot fall into the scope of usual aggregation functions, as, for instance, the (prop-erly defined) mode. We also discuss three different construction methods of suchpre-aggregation functions and we present several examples of pre-aggregationfunctions which are not aggregation functions. In particular, one of this methodis based on the usual definition of the Choquet integral on a discrete setting, but

? H. Bustince was supported by Project TIN2013-40765-P of the Spanish Government.B. Bedregal and G. Dimuro were supported by Brazilian funding agency CNPQunder Processes 481283/2013-7, 306970/2013-9, 232827/2014-1 and 307681/2012-2.

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2 M. Asiain et al.

replacing the product by other functions. The pre-aggregation functions built inthis way can be used to design a fuzzy rule-based classification system whichoutperforms some of the examples of such systems which can be found in theliterature.

Furthermore, and taking into account that for some applications such asimage processing directions along which variation data may change from onepoint ot another and may depend on the relative size of the inputs, we alsodiscuss the notion of ordered directionally monotone function.

References

1. H. Bustince, E. Barrenechea, M. Pagola, J. Fernandez Interval-valued fuzzy setsconstructed from matrices:Application to edge detection. Fuzzy Set and Systems.160 (2009) 1819–1840.

2. H. Bustince, A. Kolesarova, J. Fernandez, and R. Mesiar, Directional monotonicityof fusion functions, European Journal of Operational Research, 244 (2015) 300–308.

3. T. Calvo, A. Kolesarova, M. Komornıkova and R. Mesiar, Aggregation operators:properties, classes and construction methods. In T. Calvo, G. Mayor and R. Mesiar(Eds.): Aggregation Operators New Trends and Applications Physica-Verlag, Hei-delberg,( 2002) 3–104.

4. G. Lucca, J. Sanz, G. P. Dimuro, B. Bedregal, R. Mesiar, A. Kolesarova, and H.Bustince, Pre-aggregation functions: construction and an application, IEEE Trans-actions on Fuzzy Systems, in press, DOI: 10.1109/TFUZZ.2015.2453020.

19

A Generalized Form of a Functional EquationRelated to Scientific Laws

Juan C. Candeal

Universidad de Zaragoza. Departamento de Analisis EconomicoGran Vıa 2, 50005-Zaragoza (SPAIN).

candeal@unizar.es

Abstract. A very general functional equation which is reminiscent ofthat established for certain ‘laws of sciences’ is considered. Specifically,this equation involves partial independence with respect to ordinal scalesfor the input variables and a generalized form of an interval scale for theoutput variable. The equation is established in the setting of aggregationoperators which means that both the input and the output variables arereal-valued functions defined on an abstract space. The solution is basedupon a characterization of aggregation operators which are comparisonmeaningful with respect to independent ordinal scales, nondecreasing,and idempotent. Some other related functional equations are analyzedand an application to the theory of social choice is also shown.

References

1. Aczel J, Roberts FS, Rosembaum Z (1986) On scientific laws without dimensionalconstants. J. Math. Anal. Appl. 119:389-416.

2. Candeal JC (2015) Social evaluation functionals: a gateway to continuity in socialchoice. Soc. Choice Welf. 44:369-388.

3. Candeal JC (2016) Aggregation operators, comparison meaningfulness and socialchoice. J. Math. Psychol. (to appear) http://dx.doi.org/10.1016/j.jmp.2015.10.005

4. Chichilnisky G (1982) Social aggregation rules and continuity. Q. J. Econ. 97:337-352.

5. Grabisch M, Marichal JL, Mesiar R, Pap E (2009) Aggregation functions. Ency-clopedia of Mathematics and Its Applications 127. Cambridge University Press,Cambridge, UK.

6. Krantz DH, Luce RD, Suppes P, Tversky A (1971) Foundations of measurement.Vol. I: Additive and polynomial representations. Academic Press, New York.

7. Luce RD (1959) On the possible psychophysical laws. Psychol. Rev. 66:81-95.8. Luce RD, Raiffa H (1957) Games and Decisions: Introduction and Critical Survey.

John Wiley & Sons, Inc. New York.9. Marichal JL (2002) On order invariant synthesizing functions. J. Math. Psych.

46:661-676.10. Marichal JL, Mesiar R (2009) A complete description of comparison meaningful

aggregation functions mapping ordinal scales into an ordinal scale: a state of theart. Aequationes Math. 77(3):207-236.

11. Roberts FS (1979) Measurement theory with applications to decisionmaking, util-ity and social sciences. Addison-Wesley, Reading, MA.

20

The HOGS procedure: a sampling approach toaggregate subgroups of a poset

G. Caperna, G. Boccuzzo

University of Padova, Department of Statistical Science

Abstract

The simplification of complex and unobservable concepts is the reason of theinterest for synthetic measures in applied sciences. This interest led to differentapproaches of formalization and methodologies [1].Despite the use of synthetic indicators is widespread, many methodological is-sues arise. Especially in social science, many concepts are subjective, and areoften measured on an ordinal or dichotomous scale in large surveys that involvethousands of individuals.The moving issues, arise from specific needs due to the application to social andofficial statistics: the use of ordinal data, the implementation of profiles’ fre-quency in order to take the distribution of observed variables into account, thecomplexity of the poset structure in the case of big datasets, and the dimensionallimitation of software respect to the mean of the rank of profiles among linearextensions (average rank), and the approximation of the average rank.The HOGS procedure (Height Of Groups by Sampling) is conceived to handlea big set of observations using the results developed in the theory of partiallyordered sets, overcoming some of the usual limitations of the approaches basedon the approximation of average rank.This new procedure computes the mean of average rank of groups of units,identified respect to common external explanatory variables, and allows the in-vestigation of the relations between the ranks and the explanatory variables.The utilization of poset theory allows the management of ordinal data while ourmethod adapts it to the case of big datasets characterized by complex distribu-tions.The procedure consists in the observation of sub-samples of the population. Inevery sub-sample the average rank of every vertex (also called profile in its ap-plication to statistics where they are defined by the vectors of observed data)is approximated [2,3], with methods based on the dimension of the down/upsets and the set of incomparables as defined in [4]. Then the elements of everysub-sample are stratified by an external variable (gender, region, . . . ), and thenaggregated computing the mean of average ranks in every stratum. We applythis method to study the relation of socio-economic conditions to life satisfaction.

This work proposes specific advantages: first of all a new approach for thecomputation of an aggregated measure of an unobservable concept observed onmultiple ordinal (or mixed) variables, secondly a procedure to study the effect of

21

explanatory (external) variables on the different levels of such a concept. Finally,the HOGS procedure allows to handle big data in small portions, offering asolution to take into account the frequency distribution of data and reducingthe issues determined by the dimension of social surveys, often made by tens ofthousands of observations.

References

1. Nardo, M., Saisana, M., Saltelli, A., Tarantola, S., Hoffman, A., Giovannini, E.:Handbook on constructing composite indicators. OECD Statistic Working Paper(2005)

2. Brüggemann, R., Carlsen, L.: An improved estimation of averaged ranks of partialorders. MATCH Commun.Math.Comput.Chem 65 (2011) 383–414

3. De Loof, K., De Baets, B., De Meyer, H.: Approximation of average ranks in posets.MATCH- Commun.Math.Comput.Chem. 66 (2011) 219–229

4. Brüggemann, R., Patil, G.P.: Ranking and Prioritization for Multi-indicator Sys-tems. Volume 5. Springer, New York (2011)

22

Preferences and aggregation operatorsover property spaces

Marta Cardin

Department of EconomicsUniversita Ca’ Foscari Venezia

Decision problems are characterized by a plurality of points of view. We have toconsider the different dimensions from which the alternatives can be viewed ina multi-attribute decision model or the preferences of voters in a social choiceproblem.In order to solve a decision problem we have to compare and rank a set of al-ternatives. In this note we consider the model of abstract Arrowian aggregationintroduced in [10] that represents a decision problem in term of a set of Booleanproperties specifying for every alternative a list of properties that are satisfied.A property space is a pair (X;H) where X is a non-empty set and H is a collec-tion of non-empty subsets of X and if x, y ∈ X and x 6= y there exists H ∈ Hsuch that x ∈ H and y /∈ H. The elements of H are referred to as properties andif x ∈ H we say that x has property represented by the subset H. Our definitionis slightly more general than that of [10] and we do not assume that the set Xis finite as in [10].The “property space” model has received attention in the literature on judge-ment aggregation for studying the problem of aggregating sets of logically inter-connected propositions. Moreover it provides a general framework for represent-ing preferences and then aggregation of preferences.We prove that every property space defines a lattice structure on the set X andalso that every distributive lattice is a property space characterized by the setH of prime filters.Then we can provide an axiomatic characterization of property space structure.We then focus on aggregation operators f : Xn → X where (X;H) is a prop-erty space and we consider operators that are componentwise compatible withthe structure of property space of X. We study compatible aggregation opera-tors that satisfy properties of monotonicity and independence and we obtain acharacterization of Sugeno integral in the framework of property spaces.

References

1. Couceiro M., Marichal J.-L. : Polynomial functions over bounded distributive lat-tices, Journal of Mul-Valued Log S, 18, 247256, (2012)

2. Couceiro M., Marichal J.L.: Characterizations of discrete Sugeno integrals as latticepolynomial functions, Proceedings of the 30th Linz Seminar on Fuzzy Set Theory(LINZ2009), 17–20, (2009)

23

2 Preferences and aggregation operators over property spaces

3. Couceiro M., Marichal J.L.: Characterizations of discrete Sugeno integrals as poly-nomial functions over distributive lattices. Fuzzy Set Syst, 161, 694–707, (2010)

4. Chambers C. P., Miller A. D.: Scholarly influence. J Econ Theory, 151(1), 571–583,(2014)

5. Chambers C.P., Miller A.D.: Benchmarking, working paper, (2015)6. Caspard N.,Leclerc B., Monjardet B.: Finite ordered sets, Encyclopedia of Math-

ematics and its applications, Cambridge University Press, (2012)7. Grabisch M. .Marichal J.L, Mesiar R., Pap E., Aggregation Functions, Encyclope-

dia of Mathematics and its Applications. Cambridge University Press, Cambridge,(2009)

8. Gratzer G., General Lattice Theory, Birkhauser Verlag, Berlin, (2003)9. Halas R., Mesiar R., Pocs J., A new characterization of the discrete Sugeno integral,

Inform Fusion, 29, 84–86, (2016)10. Nehering K., Puppe C.: Abstract Arrowian aggregation. J Econ Theory, 145, 467–

494, (2010)

24

On a New Class of Uninormson Bounded Lattices

Gul Deniz Caylı1, Funda Karacal2, and Radko Mesiar3 ⋆

1 Department of Mathematics, Faculty of Sciences, Karadeniz Technical University,61080 Trabzon, Turkey

E-mail: guldeniz.cayli@ktu.edu.tr2 Department of Mathematics, Faculty of Sciences, Karadeniz Technical University,

61080 Trabzon, TurkeyE-mail: fkaracal@yahoo.com

3 Department of Mathematics and Descriptive Geometry, Faculty of CivilEngineering, Slovak University of Technology, Radlinskeho 11, 81005 Bratislava,

SlovakiaE-mail: mesiar@math.sk

Abstract. We study and propose some new construction methods toobtain uninorms on bounded lattices. Considering an arbitrary boundedlattice L, we show the existence of idempotent uninorms on L for anyelement e ∈ L\{0, 1} playing the role of a neutral element. By our con-struction method, we obtain the smallest idempotent uninorm and thegreatest idempotent uninorm with the neutral element e ∈ L\{0, 1}. Wesee that the obtained uninorms are conjunctive and disjunctive uninorms,respectively. On the other hand, if L is not a chain, we also provide anexample of an idempotent uninorm which is neither conjunctive nor dis-junctive.

References

1. Birkhoff, G.: Lattice Theory. American Mathematical Society Colloquium Pub-lishers Providence RI, 1967.

2. De Baets, B.: Idempotent uninorms. European J. Oper. Res. 118, 631-642(1999)

3. Bodjanova, S., Kalina, M.: Construction of uninorms on bounded lattices, IEEE12th International Symposium on Intelligent Systems and Informatics, SISY2014, September 11-13, 2014, Subotica, Serbia.

4. Deschrijver, G.: Uninorms which are neither conjunctive nor disjunctive ininterval-valued fuzzy set theory. Information Sciences 244, 48–59 (2013)

5. Drygas, P.: On properties of uninorms with underlying t-norm and t-conormgiven as ordinal sums. Fuzzy Sets and Systems 161, 149–157 (2010)

6. Drygas, P., Ruiz-Aguilera, D., Torrens, J.: A characterization of uninorms lo-cally internal in A(e) with continuous underlying operators. Fuzzy Sets andSystems 287, 137–153 (2016)

⋆ Acknowledgement The third author was supported by grant APVV-14-0013.

25

7. Fodor, J., Yager, R. R., Rybalov, A.: Structure of uninorms. Internat J. Un-certain Fuzziness Knowledge-Based Systems 5, 411–427 (1997)

8. Karacal, F.: On the direct decomposability of strong negations and S-implication operators on product lattices. Information Sciences 176, 3011–3025(2016)

9. Karacal, F., Mesiar, R.: Uninorms on bounded lattices. Fuzzy Sets and Systems261, 33–43 (2015)

10. Kesicioglu, M. N., Karacal, F., Mesiar, R.: Order-equivalent triangular norms.Fuzzy Sets and Systems 268, 59–71 (2015)

11. Klement, E. P., Mesiar, R., Pap, E.: On the relationship of associative com-pensatory operation to triangular norms and conorms. Internat J.UncertainFuzziness Knowledge-Based Systems 4, 129–144 (1996)

12. Martin, J., Mayor, G., Torrens, J.: On locally internal monotonic operations.Fuzzy Sets and Systems 137, 27–42 (2003)

13. Mesiar, R., Pap, E.: Different interpretations of triangular norms and relatedoperations. Fuzzy Sets and Systems 96, 183–189 (1998)

14. Mesiarova-Zemankova, A.: Multi-polar t-conorms and uninorms. InformationSciences 301, 227–240 (2015)

15. Yager, R. R.: Misrepresentations and challenges: a response to Elkan, IEEEExpert August 41–42 (1994)

16. Yager, R. R., Rybalov, A.: Uninorms aggregation operators. Fuzzy Sets andSystems 80, 111–120 (1996)

26

Quasideviation means in insurance

Jacek Chudziak

Faculty of Mathematics and Natural Sciences, University of Rzeszow,ul. Prof. St. Pigonia 1, 35-310 Rzeszow, Poland

Abstract. Let I ⊆ IR be an open interval. A function D : I × I → IRis said to be a quasideviation provided it satisfies the following threeconditions:(D1) for every x, y ∈ I, D(x, y) is of the same sign as x− y;(D2) for every x ∈ I, the function I 3 t→ D(x, t) ∈ IR, is continuous;(D3) for every x, y ∈ I such that x < y, the function

(x, y) 3 t→ D(y, t)

D(x, t)∈ IR,

is strictly increasing.The notion of a quasideviation has been introduced by Zs. Pales [2].Quasideviations are generalizations of deviations, considered earlier byZ. Daroczy [1]. In [3] it has been proved that if D : I × I → IR is aquasideviation, then for every n ∈ IN, x1, ..., xn ∈ I and λ1, ..., λn ∈[0,∞) with

∑n

i=1λi > 0, equation

n∑i=1

λiD(xi, t) = 0 (1)

has a unique solution t0 ∈ [min{xi : i ∈ {1, .., n}},max{xi : i ∈ {1, .., n}}].In this way, equation (1) defines a mean, called a quasideviation meanof x1, ..., xn weighted by λ1, ..., λn. It turns out that some quasideviationmeans are closely related to an important notion of insurance mathe-matics, namely the zero utility principle. Applying the results in [3], weprove several properties of that principle.

References

1. Daroczy Z.,Uber eine Klasse von Mittelwerten, Publ. Math. Debre-cen 19 (1972), 211–217.

2. Pales Zs., Characterization of quasideviation means, Acta. Math.Sci. Hungar. 40 (1982), 243–260.

3. Pales Zs., General inequalities for quasideviation means, Aequa-tiones Math. 36 (1988), 32–56.

27

Using aggregation functions on structured data:a use case in the FIGHT-HF project

Kevin Dalleau1, Miguel Couceiro1, Marie-Dominique Devignes1, Chedy Raissi1,and Malika Smail-Tabbone1

LORIA (CNRS, Inria NGE, University of Lorraine), Nancy, Francekevin.dalleau@loria.fr

Abstract. Heart Failure is a serious condition, affecting more and morepeople every year. The project FIGHT-HF aims at getting a better un-derstanding of the disease and finding innovative ways to prevent andtreat it. In order to enable an evidence-based decision making and de-sign novel therapeutic strategies from the collected data, efficient dataintegration and mining tools must be used. We propose a method toaggregate complex graphs into simple weighted graphs, and to apply iton these data, containing complex relations between entities of variousnature.

Keywords: Graph aggregation, Complex graphs, Clustering, Heart Fail-ure

Heart Failure (HF) is a pandemic heart disease and a major public healthissue, particularly on elderly people. The ambition of ”FIGHT-HF”, a Frenchnational research program coordinated by Prof. P. Rossignol (Regional Univer-sity Hospital, Nancy, France), is to engage the battle against heart failure, bygetting a better understanding of the origins of the disease, identifying and val-idating new biotargets, improving the current classification (nosography), andfinding novel ways to prevent and treat the disease. A lot of data are availableand will be generated, leading to the need for cutting-edge data integration andmining methods and tools.

We have at hand complex and heterogeneous data. The data include, amongother, data regarding sociodemographical aspects of patients, biological and clin-ical features, drugs taken, and genetic profile. Once integrated, the data areschemaless, as the patient follow-up and measured variables are different fromone data source to another. Consequently, the patient data will be representedas graphs, due to their capacity to represent schema-less data along with theirrelationships with background domain knowledge (protein functions and interac-tions, biological pathways, etc.). Indeed, taking advantage of the domain knowl-edge, new relations can be added. An example of how using this knowledge mightbe useful in connecting data (before aggregation) is presented fig. 1. The globalgraph can be viewed as a complex graph where a subset of vertices contains thepatients nodes and other vertices correspond to attributes nodes. The first step

28

2

of this work consisted in designing an aggregation method, to aggregate suchcomplex graphs into a single graph containing only patient nodes, connectedwith weighted edges according to their similarity. This is a case of data fusion,where data of heterogeneous types are aggregated into one single graph.

Fig. 1. A novel connection between patients is added, taking advantage of the domainknowledge

The application of this aggregation method on patient enables the use ofspecific clustering algorithms on the data, and find groups of patients. Eachcluster should correspond to a subgroup of patients sharing the same form ofheart failure, thus requiring a specific care strategy.

Transforming multiple patient data graphs (corresponding to multiple points ofview) into more synthetic ones may lead to the proposal of new classifications ofthe considered disease as well as a consensual classification, using the principlesof consensus theories [1].

References

1. Olivier Hudry and Bernard Monjardet. Consensus theories: an oriented survey.Math. Sci. hum / Mathematics and Social Sciences 190/2, June.

29

Ranking rules characterized by means ofmonometrics and consensus states

Bernard De Baets and Raul Perez-Fernandez

KERMIT, Department of Mathematical Modelling, Statistics and Bioinformatics,Ghent University, Coupure links 653, B-9000 Gent, Belgium{bernard.debaets,raul.perezfernandez}@ugent.be

Abstract. Social choice theory is the science studying what conclusionscan be drawn from the preferences expressed by several voters over aset of candidates. Here, we consider the problem of ranking candidates,meaning we are dealing with the social choice subdiscipline of rankingrules. Given a set of preferences provided in the form of a profile of rank-ings (list of rankings), ranking rules obtain the ranking that best fitsthe profile. Many ranking rules have been proposed since the eighteenthcentury, when the works of Rousseau [15], Borda [2] and Condorcet [4]laid the foundations of social choice theory and, for some profiles, de-pending on the choice of ranking rule, one can obtain a different rankingas output. It will be no surprise then that there is no single absolute anduniversal ‘best’ ranking rule, as Arrow proved in [1]. Arrow’s Theoremstates that there is no ranking rule simultaneously satisfying a set ofproperties (non-dictatorship, unanimity and independence of irrelevantalternatives) that can be considered natural and desirable. Rather, thechoice of a ‘correct’ ranking rule depends on the nature of the problemand which natural or desirable properties are prioritized. Some of themost well-known ranking rules in social choice theory are plurality [16],the Borda count [2] and the Kemeny rule [5].

However, not only the choice of the ‘correct’ ranking rule is importantbut also the notion of consensus state (or simply consensus). In general,a profile is said to be in a consensus state when determining a winningranking is obvious. A consensus state can be seen as the domain of a par-tially defined ranking rule on the set of candidates. A trivial consensusstate is unanimity [5], where each voter has the exact same preferences onthe set of candidates. Another slightly more involved one is the existenceof a Condorcet ranking [4], which is a ranking where every candidate ispreferred by more than half of the voters to all the candidates rankedafter him/her. Several authors such as Nitzan [10], Lerer and Nitzan [7],Campbell and Nitzan [3] and Meskanen and Nurmi [8,9] have advocatedthat ranking rules can be characterized by a consensus state and a dis-tance function. In a recent paper [13], we stated that distance functionsare actually too restrictive, and we introduced monometrics, a new typeof functions that better fits with the nature of the problem. Like a dis-tance function, a monometric satisfies the axioms of non-negativity andcoincidence, but a monometric requires compatibility with a betweennessrelation [11] and does not impose symmetry nor the triangle inequality.

30

The introduction of a betweenness relation determines a notion of close-ness that needs to be preserved. The betweenness relation defined byKemeny [5] seems to be the most natural one in social choice theory.The best-known ranking rule that can be characterized as minimizingthe distance to a consensus state for some appropriate monometric is theKemeny rule [5], where the search for the profile of rankings that is theclosest to becoming unanimous is addressed considering a function thatis both a distance function and a monometric w.r.t. Kemeny’s betweenessrelation [5]: the Kendall (tau) distance [6]. As another example of a recentsuch ranking rule, Rademaker and De Baets proposed in [14] a rankingrule that amounts to finding the ranking for which it holds that the votesare closest to satisfying a natural property: monotonicity. For a rankinga � b � c, monotonicity means that the number of voters preferring ato c should not be less than both the number of voters preferring a to band the number of voters preferring b to c. In [12], this ranking rule wasaddressed considering monotonicity of the votrix [4,17] as the consensusstate and proposing the use of monometrics to measure the distance tosuch consensus state. As discussed in [13], monometrics and consensusstates play a key role in social choice theory characterizing most rankingrules.

References

1. K. J. Arrow, A difficulty in the concept of social welfare, The Journal of PoliticalEconomy 58 (4) (1950) 328–346.

2. J. C. Borda, Memoire sur les Elections au Scrutin, Histoire de l’Academie Royaledes Sciences, Paris, 1781.

3. D. E. Campbell, S. I. Nitzan, Social compromise and social metrics, Social Choiceand Welfare 3 (1986) 1–16.

4. M. Condorcet, Essai sur l’Application de l’Analyse a la Probabilite des DecisionsRendues a la Pluralite des Voix, De l’Imprimerie Royale, Paris, 1785.

5. J. G. Kemeny, Mathematics without numbers, Daedalus 88 (4) (1959) 577–591.6. M. G. Kendall, A new measure of rank correlation, Biometrika 30 (1938) 81–93.7. E. Lerer, S. Nitzan, Some general results on the metric rationalization for social

decision rules, Journal of Economic Theory 37 (1985) 191–201.8. T. Meskanen, H. Nurmi, Distance from consensus: A theme and variations, in:

B. Simeone, F. Pukelsheim (eds.), Mathematics and Democracy. Recent Advancesin Voting Systems and Collective Choice, Springer, Berlin, 2006, pp. 117–132.

9. T. Meskanen, H. Nurmi, Closeness counts in social choice, in: M. Braham, F. Steffen(eds.), Power, Freedom, and Voting, Springer, Heidelberg, 2008, pp. 289–306.

10. S. Nitzan, Some measures of closeness to unanimity and their implications, Theoryand Decision 13 (1981) 129–138.

11. M. Pasch, Vorlesungen uber neuere Geometrie, vol. 23, Teubner, Leipzig, Berlin,1882.

12. R. Perez-Fernandez, M. Rademaker, P. Alonso, I. Dıaz, S. Montes, B. De Baets,Representations of votes facilitating monotonicity-based ranking rules: From votrixto votex, International Journal of Approximate Reasoning 73 (2016) 87–107.

13. R. Perez-Fernandez, M. Rademaker, B. De Baets, Monometrics and their role inthe rationalisation of ranking rules, Information Fusion 34 (2017) 16–27.

31

14. M. Rademaker, B. De Baets, A ranking procedure based on a natural monotonicityconstraint, Information Fusion 17 (2014) 74–82.

15. J. J. Rousseau, Du Contrat Social, Marc Michel Rey, Amsterdam, 1762.16. E. S. Staveley, Greek and Roman Voting and Elections, Cornell University Press,

New York, 1972.17. H. P. Young, Condorcet’s theory of voting, American Political Science Review

82 (4) (1988) 1231–1244.

32

Different Types of De Morgan Identities inNilpotent Systems and a Universal Description

J. Dombi1 and O. Csiszar2

1 University of Szegeddombi@inf.u-szeged.hu

2 Obuda University, Institute of Informatics & Applied Mathematicscsiszar.orsolya@nik.uni-obuda.hu

In human thinking, averaging operators, where a high input can for a lowerone compensate, play a significant role. The aggregative operator was first in-troduced in 1982 by Dombi, by selecting a set of minimal concepts that must befulfilled by an evaluation-like operator. The concept of uninorms was introducedby Yager and Rybalov, as a generalization of both t-norms and t-conorms. Byadjusting its neutral element, a uninorm is a t-norm if ν∗ = 1 and a t-conormif ν∗ = 0. Uninorms turned out to be useful in many fields like expert systems,aggregation and fuzzy integral. The main difference in the definition of the uni-norms and aggregative operators is that the self-duality requirement does notappear in uninorms, and the neutral element property is not in the definition forthe aggregative operators. Now we distinguish between logical operators (withclassical logical values on the boundaries, e.g. conjunction, disjunction, impli-cation) and multicriteria decision tools (e.g. means, preferences) and here weconsider multicriteria decision tools. Our main purpose is to consider gener-ated nilpotent operators in an integrative frame and to examine the nilpotentself-dual generated operators. A general parametric framework for the nilpotentconjunctive, disjunctive, aggregative and negation operators is given and it isshowed, how the nilpotent generated operator can be used for preference mod-elling. First we show that by shifting the generator function of a disjunction, wecan get a conjunction and also operators that fulfil the self De Morgan property.We provide a general parametric formula for these operators, in which the con-junction, disjunction and the so-called aggregative operator differ only in onesingle parameter. This parameter has the semantical meaning of the level ofexpectancy.

Definition 1. Let f : [0, 1] → [0, 1] be a strictly increasing bijection, ν ∈ [0, 1],and x = (x1, . . . , xn), where xi ∈ [0, 1] and let us define the general operator by

oν(x) = f−1

�n�

i=1

(f(xi)− f(ν)) + f(ν)

�= f−1

�n�

i=1

f(xi)− (n− 1)f(ν)

�.

(1)

A more general, weighted form of this operator is also examined.

33

Definition 2. Let w = (w1, . . . , wn), wi > 0 real parameters, f : [0, 1] → [0, 1] astrictly increasing bijection, ν ∈ [0, 1]. The weighted generated operator is definedby

aν,w(x) := f−1

�n�

i=1

wi(f(xi)− f(ν)) + f(ν)

�. (2)

We examine the followiing question: for which parameter values satisfy theabove-defined general operator for the De Morgan property with respect to thenegation generated by f(x). A commutative weighted generated operator fulfilsthe self De Morgan property if and only if w = 1

n or ν = ν∗, where f(ν∗) = 12 ;

i.e. it has one of the following forms:

f−1

�1

n

n�

i=1

f(xi)

�(3)

or

f−1

�w

�n�

i=1

f(xi)−n

2

�+

1

2

�. (4)

The weighted generated operator of the form f−1

�w

�n�

i=1

f(xi)− n2

�+ 1

2

�,

is commutative and satisfies the self De Morgan property. For its nice propertiesit is sensible to give it a distinctive name. The operator

aw(x) = f−1

�w

�n�

i=1

f(xi)−n

2

�+

1

2

�, (5)

where w > 0 is called weighted aggregative operator.An important property of aggregation functions concerns the grouping char-

acter; i.e. whether it is possible to build a partial aggregation for subgroups ofinput values, and then to get the overall value by combining these partial re-sults. We show that the weighted aggregative operator with weights w ≤ 1

n isbisymmetric.

34

(a) w = 0.5 (b) w = 0.75

(c) w = 1

Fig. 1. The weighted aggregative operator aw for f(x) =1

1 + νd1−νd

1−xx

, νd = 0.8

We thoroughly examine the weighted aggregative operator of two variables.We show that a1(x, y) has a uninorm-like property and it satisfies the selfDe Morgan property as well. However, it is not associative (since a1(0, 1) =a1(1, 0) = f−1

�12

�= ν∗), and therefore cannot be a uninorm.

By substituting n(x) and y in the commutative self De Morgan weightedaggregative operator, the operator a(n(x), y) has certain properties which aresimilar to those expected of a preference operator.

35

Properties of uninorms and their generalization

Pawe l Drygas

Faculty of Mathematics and Natural Sciences, University of Rzeszow35-959 Rzeszow, ul. Pigonia 1, Poland

paweldrs@ur.edu.pl

Uninorms were introduced by Yager and Rybalov in 1996 [8]. They are im-portant generalizations of triangular norms and conorms because they allow aneutral element to lie anywhere in the unit interval rather than at zero or oneas in the case of t-norm and t-conorm.

Fodor, Yager and Rybalov examined the general structure of uninorms, forexample, the frame structure of uninorms and characterization of representableuninorms are presented in [3]. In the next papers we can find other properties(see [1, 2, 7]).

In this paper we present some properties of increasing, associative binaryoperations in the unit interval with a neutral element e ∈ (0, 1). More specifically,we ask which of the property of uninorms will be preserved, if we omit some ofthe conditions in the definition of uninorms. We will deal with, among others,properties related to associativity, commutativity or a neutral element. Thismeans that they will be discussed and compare certain properties of uninorms,week uninorms, semi-uninorms, pseudo-uninorms (see [8, 4, 6]) and summarizedthe relationship between different groups of assumptions from the definition ofuninorm and its properties.

References

1. Drewniak, J., Drygas, P.: Ordered semigroups in constructions of uninorms andnullnorms. in: P. Grzegorzewski, M. Krawczak, S. Zadrozny (Eds), Issues in SoftComputing. Theory and Applications, EXIT, Warszawa, 2005, 147 - 158

2. Drygas, P., Ruiz-Aguilera, D., Torrens, J: A characterization of a class of uninormswith continuous underlying operators. Fuzzy Sets Syst. 287, 137-153 (2016)

3. Fodor, J.C., Yager, R.R., Rybalov A.: Structure of uninorms. Internat. J. Uncertain.Fuzziness Knowledge-Based Syst. 5, 411–427 (1997)

4. Jiachao, W., Maokang, L.: Some properties of weak uninorms. Information Sciences,181, 3917-3924 (2011)

5. Klement, E.P,. Mesiar, R., Pap, E.: Triangular norms, Kluwer Acad. Publ., Dor-drecht, 2000

6. Liu, H.-W.: Semi-uninorms and implications on a complete lattice. Fuzzy Sets Syst.,191, 7282 (2012)

7. Ruiz-Aguilera, D., Torrens, J.,De Baets, B., Fodor, J.C.: Some Remarks on theCharacterization of Idempotent Uninorms. in: E. Hllermeier, R. Kruse, F. Hoffmann(Eds.), Proceedings of the 13th IPMU Conference 2010, in: Lect. Notes Comput. Sci.,vol. 6178, Springer-Verlag, 425434 (2010)

8. Yager, R.R., Rybalov, A.: Uninorm aggregation operators. Fuzzy Sets Syst. 80,111–120 (1996)

36

Aggregation in logical consequence: revisiting fromgraded context

Soma Dutta§, Mihir K. Chakraborty?, and Andrzej Skowron∗

§Vistula University, Warsaw, Poland∗,§Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland

?Jadavpur University, Kolkata, Indiasomadutta9@gmail.com mihirc4@gmail.com skowron@mimuw.edu.pl

Use of aggregation [1] has different purposes, and accordingly different treatments.Statistical nature of a data requires to be handled with typical statistical tools of ag-gregation, e.g., average, weighted average etcetera. Information expressed in terms offinitely many sentences of a language, or attributes of a concept, is aggregated usingconnectives like conjunction, disjunction. In case of consequence in logic, a formulaα follows from a set of formulae X if for all possible valuation function Ti’s if Ti is amodel of X (i.e., every member of X is true under Ti or Ti satisfies every member of X),then α is also true under Ti (or Ti satisfies α). That is, in order to come to a decision thatwhether α follows from X , one needs to first check whether for a Ti, X is satisfied by Tiimplies α is satisfied by Ti, and then aggregate the case for all possible Ti. Intuitively, ifwe consider each Ti as a source or agent or expert, who has different opinion regardingwhether if every member of X is true then α is true, then the quantifier ‘all’ works as anaggregation operation accumulating the opinion of each Ti.

In this presentation we would concentrate on the nature of aggregation, which isrequired to have a faithful representation of the notion of logical consequence. In thisregard, we shall pass on from the classical context of consequence to a graded context,as to learn the general nature of such an aggregation operation, instead of two values,dealing with many values would be more insighful.

Classically, the notion of semantic consequence, denoted as |=, is defined as X |= α

iff for all possible valuation functions Ti : F 7→ {0,1}, from the set of all formulas to thevalue set {0,1}, if every member of X receives the value 1 (true) then α also receivesthe same. Identifying Ti with a set, consisting of those formulas from F which receives1 under the function Ti, X |= α turns out to be ∀Ti{∀(x ∈ X → x ∈ Ti)→ α ∈ Ti} . . . (Σ).

That a formula is a logical consequence of a set of formulas, involves two meta-level operations; one is universal quantification (∀), and the other is implication (→).The above definition, given by (Σ), went through a generalization when Shoesmith andSmiley [4] proposed to consider a collection of {Ti}i∈I instead of all possible valuationfunctions, and the notion X |= α is relativized by X |={Ti}i∈I α. Based on that generaliza-tion, next generalization appears when instead of a collection of two-valued functions,{Ti}i∈I is considered to be a collection of fuzzy sets or functions from F to a general(complete) lattice structure L bounded by the top (1) and the least (0). That is, givenany formula α, α ∈ Ti is now having a value, possibly other than the top or the least,in L. So, both the right hand side expression (α ∈ Ti) and the left hand side expression(∀x(x ∈ X → x ∈ Ti)) of→ present in (Σ) get some values in L. That is, α follows fromX , which may be denoted as X |≈{Ti}i∈I α, is now a many-valued notion. So X |≈{Ti}i∈I α

37

has a value, denoted as gr(X |≈{Ti}i∈I α), in L. The value gr(X |≈{Ti}i∈I α) is supposedto be the value of the sentence (Σ) with respect to the value set L endowed with somealgebraic operation for the connective→ present in (Σ).

As we go beyond the two-valued set up, operations for computing ∀ and→ are re-quired. Like the algebraic semantics of classical logic is captured in a Boolean algebra,and that of intuitionistic logic is obtained in a Heyting algebra, a complete residuatedlattice structure (L,∗m,→m,0,1) is required to have a sound interpretation for a notionof graded consequence, which is a fuzzy relation |∼ from the set of all subsets of for-mulae (P(F )) to F , satisying following conditions.(GC1) If α ∈ X , then gr(X |∼ α) = 1.(GC2) If X ⊆ Y , then gr(X |∼ α)≤ gr(Y |∼ α).(GC3) infβ∈Y gr(X |∼ β)∗m gr(X ∪Y |∼ α)≤ gr(X |∼ α).

These are respectively the generalization of the properties, namely overlap, dilu-tion, and cut of a classical consequence relation in the context of graded consequencerelation. With respect to the value set L endowed with the above mentioned algebraicstructure, the value of the expression (Σ), in graded context, is computed (algebraically)using the operators ‘inf’ and ‘→m’ for ∀ and→ respectively, and it becomes gr(X |≈ α)= infi∈I{infx∈X Ti(x)→m Ti(α)} . . . (Σ′).

The theory of graded consequence, based on the above set up, is developed to aconsiderable length [2, 3]. Here, we take an attempt to look back some of the draw-backs of the way of aggregation, proposed in (Σ′), from a practical perspective. The‘infimum’ operation of a complete lattice is considered to be the algebraic translationof the linguistic quantifier ∀. Sometimes, ∀ is considered to be a generalization of con-junctive aggregation operations, which are algebraically represented by t-norms. Infi-mum (inf) is a special kind of t-norm having the property of idempotence, and is thegreatest among all t-norms. From the practical perspective, if Ti’s are considered to bethe experts, who evaluate whether α follows from X (in terms of assigning values to(∀x(x ∈ X → x ∈ Ti)→ α ∈ Ti) from L), then while aggregating all the experts’ opinionusing infimum, it boils down to the lowest; even if that lowest value does not appear assomeone’s opinion, or appears as an isolated value among a majority of greater-valuedopinions, the lowest prevails in the final aggregation. So, aggregation using ‘inf’ oftendoes not fit well in practical context. We hence, in this presentation, look for a differentnotion of aggregation which can be faithful to the theoretical interpretation of |=, andtakes care of the practicality of aggregation too.

References

1. Beliakov G., Pradera A., Calvo T., Aggregation Functions: A Guide for Practitioners, InStudies in Fuzziness and Soft Computing, Volume 221, Springer 2007.

2. Chakraborty M.K., Graded Consequence: further studies, Journal of Applied Non-ClassicalLogics, 5(2), 127-137, 1995.

3. Dutta S., Bedregal B., Chakraborty M.K., Some Instances of Graded Consequence in theContext of Interval-Valued Semantics. Banerjee and Krishna (Eds.): ICLA 2015, LNCS8923, 74-87 (2015)

4. Shoesmith, D.J. and Smiley, T.J., Multiple Conclusion Logic. Cambridge University Press(1978)

38

A Group Decision Making Procedure in theContext of Non-Uniform Qualitative Scales

Jose Luis Garcıa-Lapresta1 and David Perez-Roman2

1 PRESAD Research Group, BORDA Research Unit, IMUVA, Dept. de EconomıaAplicada, Universidad de Valladolid, Spain

2 PRESAD Research Group, BORDA Research Unit, Dep. de Organizacion deEmpresas y C.I.M., Universidad de Valladolid, Spain

In this paper, we introduce a new voting system in the context of orderedqualitative scales. The process is conducted in a purely ordinal way by consid-ering an ordinal proximity measure that assigns an ordinal degree of proximityto each pair of linguistic terms of the qualitative scale. Once the agents assessthe alternatives through the qualitative scale, the alternatives are ranked ac-cording to the medians of the ordinal degrees of proximity between the obtainedindividual assessments and the highest linguistic term of the scale. Since somealternatives may share the same median, a tie-breaking procedure is introduced;it is based on an appropriate linear order on the set of feasible medians. Someproperties of the proposed voting system have been provided.

We first recall the notion of ordinal proximity between linguistic terms withvalues on a finite chain (linear order), introduced by Garcıa-Lapresta and Perez-Roman [GL-PR]. Consider an ordered qualitative scale L = {l1, . . . , lg} suchthat l1 < · · · < lg and g ≥ 3, whose elements are linguistic terms, and a chain∆ = {δ1, . . . , δh}, with δ1 � · · · � δh. The elements of ∆ have no meaning andthey only represent different degrees of proximity, being δ1 and δh the maximumand minimum degrees, respectively.

Definition 1. ([GL-PR]) An ordinal proximity measure on L with values in ∆is a mapping π : L2 −→ ∆, where π(lr, ls) = πrs means the degree of proximitybetween lr and ls, satisfying the following conditions:

1. Exhaustiveness: For every δ ∈ ∆, there exist lr, ls ∈ L such that δ = πrs.2. Symmetry : πsr = πrs, for all r, s ∈ {1, . . . , g}.3. Maximum proximity : πrs = δ1 ⇔ r = s, for all r, s ∈ {1, . . . , g}.4. Monotonicity : πrs � πrt and πst � πrt, for all r, s, t ∈ {1, . . . , g} such thatr < s < t.

Consider a set of agents A = {1, . . . ,m}, with m ≥ 2, that have to evaluate aset of alternatives X = {x1, . . . , xn}, with n ≥ 2, through an ordered qualitativescale L = {l1, . . . , lg}, l1 < · · · < lg, with g ≥ 3, and an ordinal proximitymeasure π : L2 −→ ∆. The agents’ judgments on the alternatives are collectedin a profile, that is a matrix V = (vai ) consisting of m rows and n columns oflinguistic terms, where the element vai ∈ L represents the linguistic assessmentgiven by the agent a ∈ A to the alternative xi ∈ X.

For ranking the alternatives, the procedure is divided in the following steps.

39

2

1. For each alternative xi ∈ X, consider the assessments obtained by xi for allthe agents: v1i , . . . , v

mi ∈ L (column i of the profile).

2. For each alternative xi ∈ X, calculate the ordinal proximities between theassessments obtained by xi and the highest linguistic term lg:

π(v1i , lg

), . . . , π (vmi , lg) ∈ ∆.

3. For each alternative xi ∈ X, arrange the previous ordinal degrees in a de-creasing fashion and select the median(s), Mi:(a) If the number of assessments is odd, then we duplicate the median. Thus,

Mi = (δr, δr) for some r ∈ {1, . . . , h}.(b) If the number of assessments is even, then we take into account the two

medians. Thus, Mi = (δr, δs) for some r, s ∈ {1, . . . , h} such that r ≤ s.Consequently, Mi ∈ ∆2, where ∆2 is the set of feasible medians:

∆2 = {(δr, δs) ∈ ∆2 | r ≤ s}.

4. For ordering the medians of ordinal proximities obtained by different alter-natives in the previous step, consider the linear order � on ∆2 definedas

(δr, δs) � (δt, δu) ⇔

r + s < t+ u

or

r + s = t+ u and s− r ≤ u− t,(1)

for all (δr, δs), (δt, δu) ∈ ∆2.5. Finally, the alternatives are ranked according to the weak order < on X

defined as xi < xj ⇔ Mi �Mj .

Since some alternatives can share the same median(s), it is necessary to de-vise a tie-breaking process for ordering the alternatives. We propose to use asequential procedure based on Balinski and Laraki [BL]. It consists of droppingthe median(s) of the respective alternatives that are in a tie, and then select thenew median(s) of the remaining ordinal degrees for the corresponding alterna-tives and applying the procedure given in (1). The process continues until the tiesare broken. It is important noticing that alternatives with different assessmentsnever are in a final tie.

We note that the devised group decision making procedure satisfies anonymity,neutrality, independence of irrelevant alternatives, unanimity, monotonicity andreplication invariance, among other properties.

References

[BL] Balinski, M., Laraki, R.: A theory of measuring, electing and ranking. Proceed-ings of the National Academy of Sciences of the United States of America 104,8720–8725 (2007)

[GL-PR] Garcıa-Lapresta, J.L., Perez-Roman, D.: Ordinal proximity measures in thecontext of unbalanced qualitative scales and some applications to consensusand clustering. Applied Soft Computing 35, 864–872 (2015)

40

Interval-valued Extended Bonferroni MeanOperator and Its Applications in Multi-criteria

Decision Making

Debashree Guha1 and Radko Mesiar2

1Indian Institute of Technology Patna, Department of Mathematics,Patna 801103, India

2Slovak University of Technology, Fac. of Civil Engineering,Radlinskeho 11, 810 05 Bratislava, Slovakia

{debashree@iitp.ac.in;radko.mesiar@stuba.sk}

In recent years, there has been a growing research interest in analyzing ex-tended Bonferroni mean (EBM) operator [5, 3] and applying it for solving deci-sion making problems. The aim of EBM operator is to capture the heterogeneousrelationship among the input data. Its use for the extension of fuzzy sets, suchas, interval valued fuzzy sets or Atanassov’s Intuitionistic fuzzy sets [4] are be-coming very popular in the literature. For many of these applications we need topresent a general method that also helps to build EBM for all environments. Forthis purpose, we first study the point-wise operations for intervals [1]. We discussthe concept of admissible orders between intervals [2] in terms of two aggrega-tion functions. Next, we present the definition of interval-valued EBM operatorsbased on a fixed admissible order and point-wise operations for intervals. Inaddition, we investigate several desirable properties of the proposed operatorsand we prove that some known specific aggregation operators are special casesof the proposed interval-valued EBM operators. The influence of the interactionamong data on the proposed EBM is analyzed by observing the variations of theaggregated value with respect to the changes of interrelationship structure. Wealso investigate the behaviour of the proposed operators by using suitable ex-amples. Then we apply the proposed operators in multi-criteria decision makingproblem.

Acknowledgment The first author kindly acknowledge the support of SAIA-NSP scholarship given by the Government of Slovak Republic. The second au-thor kindly acknowledge the support of the project of Science and TechnologyAssistance Agency under the contract No. APVV–14–0013.

References

1. Bustince, H., Galar, M., Bedregal, B., Kolesarova, A., Mesiar, R.: A New Approachto Interval-Valued Choquet Integrals and the Problem of Ordering in Interval-Valued Fuzzy Set Applications. IEEE Trans. Fuzzy Syst., 21, 1150-1162 (2013).

2. Bustince, H., Fernandez, J., Kolesarova, A., Mesiar, R.: Generation of Linear Or-ders for Intervals by means of Aggregation Functions. Fuzzy Sets and Systems,220, 69-77 (2013).

41

2 D. Guha, R. Mesiar

3. Chen, Z.S., Chin, K.S., Li, Y.L., Yang, Y.: On Generalized ExtendedBonferroni Means for Decision Making. IEEE Trans. Fuzzy Syst., DOI:10.1109/TFUZZ.2016.2540066.

4. Das, S., Guha, D., Mesiar, R.: Extended Bonferroni Mean under IntuitionisticFuzzy Environment Based on a Strict t-conorm. Working paper (2016).

5. Dutta, B., Guha, D., Mesiar, R.: A Model Based on Linguistic 2-tuples for Deal-ing with Heterogeneous Relationship among Attributes in Multi-expert DecisionMaking. IEEE Trans. Fuzzy Syst., 23, 1817-1831 (2015).

42

Lattices with a smallest set of aggregation

functions

Radomır Halas1,⋆, Jan Kuhr1,⋆⋆, and Jozef Pocs1,2,⋆ ⋆ ⋆

1 Department of Algebra and Geometry, Faculty of Science, Palacky UniversityOlomouc, Czech Republic

2 Mathematical Institute, Slovak Academy of Sciences, Kosice, Slovakia

In a recent paper [1] we have studied aggregation functions CL on a latticeL by a clone theory approach. Recall that a clone is a set of functions closedunder projections and their composition. For any finite n-element lattice L wepresented a set of at most 2n + 2 aggregation functions on L from which theclone CL is generated.

The aim of our talk is to present a characterization of all finite lattices L

for which the clone CL is as small as possible, i.e. when it coincides with theclone of 0, 1-polynomial functions on L. Clearly, this problem is closely relatedto a well-known description of so-called order polynomially complete lattices [2].These are shown to be completely determined by their tolerances, also severalsufficient purely lattice conditions will be presented. In particular, all simplerelatively complemented lattices or simple lattices for which the join (meet) ofatoms (coatoms) is 1 (0) are of this kind.

References

1. R. Halas, J. Pocs, “On the clone of aggregation functions on bounded lattices”,Information Sciences, vol. 329, pp. 381–389, 2016.

2. K. Kaarli, A.F. Pixley, “Polynomial Completeness in Algebraic Systems”, Chap-man and Hall, 2001.

⋆ The author was supported by the international project Austrian Science Fund(FWF)-Grant Agency of the Czech Republic (GACR) no. 15-34697L.

⋆⋆ The author was supported by the international project Austrian Science Fund(FWF)-Grant Agency of the Czech Republic (GACR) no. 15-34697L.

⋆ ⋆ ⋆ The author was supported by the IGA project of the faculty of Science PalackyUniversity Olomouc no. PrF2016006 and by the Slovak VEGA Grant no. 2/0044/16.

43

Generalized comonotonicity and

a characterization of discrete Sugeno integrals

on bounded distributive lattices

Radomır Halas1,⋆, Radko Mesiar2,⋆⋆, and Jozef Pocs1,3,⋆ ⋆ ⋆

1 Department of Algebra and Geometry, Faculty of Science, Palacky UniversityOlomouc, Czech Republic. radomir.halas@upol.cz

2 Department of Mathematics and Descriptive Geometry, Faculty of CivilEngineering, Slovak University of Technology in Bratislava, Slovakia

radko.mesiar@stuba.sk3 Mathematical Institute, Slovak Academy of Sciences, Kosice, Slovakia.

pocs@saske.sk

Sugeno integral was introduced by Sugeno in [3] for fusion of informationobtained in a fuzzy set characterized by its membership function. Recall that theSugeno integral is formally introduced as the Lebesgue and Choquet integrals,replacing the standard arithmetic operations + and · on the real unit interval[0, 1] by the lattice operations supremum and infimum respectively. Consideringits disjunctive normal representation enables to extend the original definitionof the Sugeno integral to the more general case, i.e., where the interval [0, 1] isreplaced by any bounded distributive lattice L, cf. [1] and [2].

If L is a bounded chain, it is well-known that the Sugeno integral on L canbe characterized as a comonotone maxitive and min-homogeneous aggregationfunction. Our aim is to study a similar characterization for the Sugeno integralon any bounded distributive lattice L. We say that a pair x,y ∈ Ln of vectorsis called generalized comonotone if for every pair i, j ∈ {1, . . . , n} we have

(xi ∨ yi) ∧ (xj ∨ yj) = (xi ∧ xj) ∨ (yi ∧ yj).

We show that generalized comonotonicity of L-valued vectors, generalizes thenotions of comonotonicity as well as comparability of vectors in Ln. Based onthis notion an axiomatization of L-valued Sugeno integrals is introduced.

References

1. Couceiro, M., Marichal, J.-L.: Characterizations of discrete Sugeno integrals aspolynomial functions over distributive lattices, Fuzzy Sets and Systems 161, 694–707 (2010).

⋆ The author was supported by the international project Austrian Science Fund(FWF)-Grant Agency of the Czech Republic (GACR) no. 15-34697L.

⋆⋆ The author was supported by the Slovak Research and Development Agency undercontract APVV–14–0013.

⋆ ⋆ ⋆ The author was supported by the IGA project of the faculty of Science PalackyUniversity Olomouc no. PrF2015010 and by the Slovak VEGA Grant no. 2/0044/16.

44

2. Marichal, J.-L.: Weighted lattice polynomials, Discrete Mathematics 309(4), 814–820 (2009).

3. Sugeno, M.: Theory of fuzzy integrals and its applications, Ph.D. Thesis, TokyoInstitute of Technology, Tokyo, 1974.

45

1

Some Notes on Nullnorms on Bounded Lattices

Mehmet Akif Ince 1 , Funda Karacal 2 and Radko Mesiar 3

1 Recep Tayyip Erdogan UniversityDepartment of Mathematics, 53100 Rize, Turkey

E-mail: ma-ince@hotmail.com2 Karadeniz Technical University

Department of Mathematics, 61080, Trabzon, TurkeyE-mail: fkaracal@yahoo.com

3 Slovak University of TechnologyDepartment of Mathematics and Descriptive Geometry, Faculty of Civil

Engineering, Radlinskeho 11, 81 005 Bratislava, SlovakiaE-mail: mesiar@math.sk

Nullnorms are aggregation functions which are generalizations of triangularnorms and triangular conorms with a zero element in the interior of the unitinterval. In this study, we work on nullnorms which are defined on an arbi-trary bounded lattice and we obtain interesting results. We introduce a generalmedian-based method for constructing nullnorms by means of triangular normsand triangular conorms. Furthermore, we highlight a significant difference be-tween the (existence and representation of) (idempotent) nullnorms on chains,distributive bounded lattices and general bounded lattices.

1 Acknowledgement

The work on this paper was supported by grant VEGA 1/0420/15 and RecepTayyip Erdogan University, Scientific Research Coordination Unit FBA2016-603.

References

1. Asıcı, E., Karacal, F.: On the T -partial order and properties. Information Sciences267, 323–333 (2014)

2. Birkhoff, G.: Lattice Theory. American Mathematical Society Colloquium Publish-ers, Providence, RI, (1967)

3. Calvo, T., DeBaets, B., Fodor, J.: The functional equations of Frank and Alsina foruninorms and nullnorms. Fuzzy Sets and Systems 120, 385–394 (2001)

4. Demirci, M.: Aggregation operators on partially ordered sets and their categoricalfoundations. Kybernetika 42, 261–277 (2006)

5. Drewniak, J., Drygas, J., Rak, E.: Distributivity between uninorms and nullnorms.Fuzzy Sets and Systems 159, 1646–1657 (2008)

6. Drygas, P.: A characterization of idempotent nullnorms. Fuzzy Sets and Systems145, 455–461 (2004)

46

2

7. Fodor, J., C., Yager, R., R., Rybalov, A.: Structure of uninorms. Int. J. UncertaintyFuzziness Knowledge-Based Systems 5, 411–427 (1997)

8. Grabisch, M., Marichal, J-L., Mesiar, R., Pap, E.: Aggregation Functions. Cam-bridge University Press, Cambridge (2009)

9. Karacal, F., Ince, M., A., Mesiar, R.: Nullnorms on bounded lattices. InformationSciences 325, 227–236 (2015)

10. Karacal, F., Khadjiev, Dj.:∨- distributive and infinitely

∨-distributive t-norms

on complete lattice. Fuzzy Sets and Systems 151, 341–352 (2005)11. Klement, E., P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Pub-

lishers, Dordrecht (2000)12. Mas, M., Mayor, G., Torrens, J.: t-operators. Int. J. Uncertainty Fuzziness

Knowledge-Based System 7, 31–50 (1999)13. Mas, M., Mayor, G., Torrens, J.: t-operators and uninorms on a finite totally

ordered set. (Special Issue: The Mathematics of Fuzzy Sets), Internat. J. Intell.Systems 14, 909–922 (1999)

14. Mas, M., Mayor, G., Torrens, J.: The distributivity condition for uninorms andt-operators. Fuzzy Sets and Systems 128, 209–225 (2002)

15. Mas, M., Mayor, G., Torrens, J.: The modularity condition for uninorms and t-operators. Fuzzy Sets and Systems 126, 207-218 (2002)

16. Qin, F., Zhao, B.: The distributive equations for idempotent uninorms and null-norms. Fuzzy Sets and Systems 155, 446–458 (2005)

17. Xie, A., Liu, H.: On the distributivity of uninorms over nullnorms. Fuzzy Sets andSystems 211, 62–72 (2013)

47

Elicitation of fuzzy partial orders fromincomplete preferences

Simon James

School of Information Technology, Deakin University, Australiasjames@deakin.edu.au

Abstract. Recently we have proposed the framework of fuzzy partialorder based preference relations for expression and aggregation of prefer-ences. For a set of n alternatives/options, a FPO-based preference rela-tion is represented as an n×n matrix A where each entry aij representsthe degree to which option i is preferred to option j. While this kind ofrepresentation has been researched extensively, e.g. with multiplicativeand additive relations, the key difference here is that a value of aij = 1is interpreted as indicating option i is preferred to j, a value of aij = 0means that option i is not preferred to j and values in-between representpartial preference. We therefore have the restriction that aij > 0 impliesaji = 0, and the maximum expression of strength of preference is onlycrisp preference.

The perceived advantage of such a representation is that the aggregationof such matrices is less susceptible to extreme opinions, correspondingwith a fuzzy version of the Kemeny distance. It also should align morewith a natural expression of preference that is not as dependent on in-dividual interpretations of a ratings scale.

While we have developed methods for obtaining final rankings of alter-natives through aggregation and for repairing inconsistent matrices, aremaining problem is how to deal with large datasets involving manyalternatives. In these situations, the elicitation of preferences becomesquite onerous on the decision maker and, on the computation side, thenumber of corresponding partial orders becomes expensively large. Wepropose to use a subset of triplets of comparison data, i.e. rankings pro-vided between 3 alternatives, in order to obtain a final ranking of thealternatives. Our goal is to reduce the amount of information and effortrequired from the decision maker but still be able to obtain an accept-able ranking. Once the theory behind this process is developed, it canbe evaluated on human subjects in terms of ease of preference elicitationand their agreement with the final ranking.

48

A New Ordering based on Fuzzy Implications

Balasubramaniam Jayaram

Department of Mathematics,Indian Institute of Technology Hyderabad, INDIA

jbala@iith.ac.in

Through out this work, let (P,≤, 0, 1) denote a bounded lattice. Let ⊗, I :P × P → P be a t-norm and an implication on P, respectively. For definitions,please see [1], [2].

It is well-known that if I has the ordering property, viz.,

x ≤ y ⇐⇒ I(x, y) = 1 . (OP)

then one reclaims the underlying order on P.However, in [3], [4] Kesicioglu and Mesiar introduced an ordering on P based

on an implication (See [2], Definition 1.1.1) as follows:

Definition 1 (Definition 8, [3]). Let I : P × P → P be an implication on P.For x, y ∈ P we say that

x �Iy ⇐⇒ ∃` ∈ P 3 I(`, y) = x . (1)

Theorem 1 (Propositions 1 & 2, [3]). Let (P,≤, 0, 1) be a bounded latticeand let I be an implication on P. For all a, b, c ∈ P, let I satisfy the followingproperties:

I(b, I(a, c)) = I(a, I(b, c)) , (EP)I(a, b) = I(NI(b), NI(a)) , (CP)

where NI is the natural negation of I, viz., NI(a) = I(a, 0), and is involutive.

Then �I

as in (1) defines an order on P. Further, x �Iy =⇒ y ≤ x.

Firstly, it is easy to see that the greatest element in (P,≤) becomes the leastelement in (P,�

I), see Remark 1(i) in [3]. Further, the ordering of comparable

elements w.r.to �I

is the reverse of the original ordering that they have in theunderlying lattice.

In this work, given a bounded lattice P and an implication on it, taking intoaccount the mixed monotonicity of fuzzy implications, we present an alternateway of obtaining order on the underlying P, as follows:

Definition 2. Let I : P × P → P be an implication on P. For x, y ∈ P we saythat

x vIy ⇐⇒ ∃` ∈ P 3 I(`, x) = y . (2)

49

Theorem 2. Let (P,≤, 0, 1) be a bounded lattice and ⊗ be a t-norm and I animplication on P, respectively. For all a, b, c ∈ P, let I satisfy the following:

I(a⊗ b, c) = I(a, I(b, c)) , (LI)I(1, b) = b . (NP)

Then

(i) vI

as in (2) defines an order on P.(ii) Further, x v

Iy =⇒ x ≤ y.

A few interesting aspects of the above ordering are worthy of note:

⊕ Comparable elements preserve the ordering as in the underlying poset.⊕ Note that the conditions in Theorem 2 are only sufficient and not necessary.

Further, the conditions required for vI

to define an ordering are differentand possibly more lenient than the ones required for �

I. See Proposition 1

below and note that while (LI) implies (EP), in the absence of an involutiveNI it need not be true that I satisfies (CP)(NI).

⊕ Thus any implication I on P that imposes the order �I

on P can also generatethe ordering v

Ion P. If I�, Il denote the sets of all implications I on P that

impose the orders �I,v

Ion P, respectively, then I� ( Il.

⊕ Using Theorem 3, it can be shown that it is possible to obtain the originalorder on P even if I does not satisfy (OP).

Proposition 1. Let (P,≤, 0, 1) be a bounded lattice and I an implication on Psuch that NI is strong. The following are equivelent:

(i) I satisfies (CP) w.r.to NI and (EP),(ii) There exists a t-norm ⊗ on P such that I satisfies (LI) and (NP).

Theorem 3. Let I : P × P → P be an implication on P and let vI

as in (2)define an order on P. For any fixed α ∈ P, let us define the following:

(i) Iα : P→ P is the partial function Iα(γ) = I(γ, α), and(ii) Fα = {β ∈ P|α ≤ β}.

The following are equivalent:

(i) x ≤ y =⇒ x vIy.

(ii) For each α ∈ P, Fα ⊆ Ran(Iα) = {δ ∈ P|∃γ ∈ P 3 Iα(γ) = δ}.

References

1. B. D. Baets and R. Mesiar, “Triangular norms on product lattices,” Fuzzy Sets andSystems, vol. 104, no. 1, pp. 61–75, 1999.

2. M. Baczynski and B. Jayaram, Fuzzy Implications, ser. Studies in Fuzziness andSoft Computing. Berlin Heidelberg: Springer-Verlag, 2008, vol. 231.

3. M. N. Kesicioglu and R. Mesiar, “Ordering based on implications,” Inf. Sci., vol.276, pp. 377–386, 2014.

4. ——, “Corrigendum to ”Ordering based on implications” [Inform. Sci. 276(2014)377-386],” Inf. Sci., vol. 282, pp. 415–416, 2014.

50

Ordinal Sum of t-norms on Bounded Lattices

Funda Karacal1, Umit Ertugrul2, and Gul Deniz Caylı3

1 Department of Mathematics, Faculty of Sciences, Karadeniz Technical University,61080 Trabzon, Turkey

E-mail: fkaracal@yahoo.com2 Department of Mathematics, Faculty of Sciences, Karadeniz Technical University,

61080 Trabzon, TurkeyE-mail: uertugrul@ktu.edu.tr

3 Department of Mathematics, Faculty of Sciences, Karadeniz Technical University,61080 Trabzon, Turkey

E-mail: guldeniz.cayli@ktu.edu.tr

Abstract. In this study, a new method to construct a t-norm from t-norms defined on discrete bounded lattices is proposed. Underlying ideaof this method is to combine bounded lattices over bounded lattice in-dex set putting bounded lattices on indexes of bounded index set. Thisconstruction method is considered for semi-groups. Moreover this con-struction method is modified to construct t-norm from t-norms definedon sub-intervals of [0, 1]. Whether it is preserving continuity is investi-gated. Some illustrative examples are added for clarity.

References

1. Birkhoff, G.: Lattice Theory. American Mathematical Society Colloquium Pub-lishers Providence RI, (1967)

2. Clifford, A. H.: Naturally totally ordered commutative semigroups. AmericanJournal of Mathematics 76, 631-646 (1954)

3. Ertugrul, U., Karacal, F., Mesiar, R.: Modified ordinal sums of triangular normsand triangular conorms on bounded lattices. International Journal of IntelligentSystems 30, 807-817 (2015)

4. Esteva, F., Godo, L.: Monoidal t-norm based logic: towards a logic for left-continuous t-norms. Fuzzy Sets and Systems 124, 271-288 (2001)

5. Goguen, J. J. A.: The fuzzy Tychonoff theorem. J. Math. Anal. Appl. 43, 734-742 (1973)

6. Hohle, U.: Commutative, residuated SOH-monoids, in: Hohle, Klement (Eds.),Non-Classical Logics and Their Applications to Fuzzy Subsets. A Handbookof the Mathematical Foundations of Fuzzy Set Theory, Theory and DecisionLibrary Series B: Mathematical and Statistical Methods, vol. 32, Kluwer Aca-demic Publishers, Dordrecht, 53-106 (1995)

7. Hohle, U.: Probabilistische Topologien. Manuscripta Math. 26, 223-245 (1978)8. Karacal, F., Khadjiev, D.: ∨-Distributive and infinitely ∨-distributive t-norms

on complete lattices. Fuzzy Sets and Systems 151, 341-352 (2015)9. Klement E. P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Pub-

lishers, Dordrecht (2000)

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10. Medina, J.: Characterizing when an ordinal sum of t-norms is a t-norm onbounded lattices. Fuzzy Sets and Systems 202, 75-88 (2012)

11. Saminger, S.: On ordinal sums of triangular norms on bounded lattices. FuzzySets and Systems 157(10), 14031416 (2006)

12. Saminger-Platz, S., Klement, E. P., Mesiar, R.: On Extensions of TriangularNorms on Bounded Lattices. Indagationes Mathematicae 19(1), 135-150 (2008)

13. Schweizer, B., Sklar, A.: Probabilistic Metric Spaces. North-Holland, New York(1983)

14. Zhang, D.: Triangular norms on partially ordered sets. Fuzzy Sets and Systems153(2), 195-209 (2005)

52

A perturbative approach to convergencetheorems for nonlinear integrals

Jun Kawabe

Shinshu University, 4-17-1 Wakasato, Nagano 3808553, Japanjkawabe@shinshu-u.ac.jp

Abstract. In this talk we present a unified approach to convergencetheorems for nonlinear integrals that may be considered as aggregationfunctions of infinite inputs. A key tool is a perturbation of functional.

Keywords: nonadditive measure, nonlinear integral, convergence theo-rem, perturbation

1 Introduction

Aggregation functions are used for aggregating a finite or infinite number ofinputs into a single output value. In multicriteria decision making the Choquetand the Sugeno discrete integrals are typical examples of aggregation functionsof finite inputs and they are obviously continuous with respect to each of theirinputs. This property is a guarantee for certain robustness and consistency anda non chaotic behavior.

By contrast the Choquet integral Ch(µ, f) and the Sugeno integral Su(µ, f)for a measurable function f : X → [0,∞] and a nonadditive (also called mono-tone) measure µ : A → [0,∞] on a measurable space (X,A) may be consideredas aggregation functions of infinite inputs. For those aggregation integrals theircontinuity corresponds to the convergence theorem of integrals, which meansthat the limit of the integrals of a sequence of functions is the integral of thelimit function. Thus many attempts have been made to formulate the monotone,bounded, and dominated convergence theorems for nonlinear integrals such asthe Choquet, the Sipos, the Sugeno, and the Shilkret. However, to the best ofknowledge, there is no unified approach to such convergence theorems in litera-ture that are simultaneously applicable to both the Lebesgue integral as a linearintegral and the Choquet, the Sipos, the Sugeno, and the Shilkret integrals asnonlinear integrals. Thus the purpose of this talk is to present a unified approachto convergence theorems for such linear and nonlinear integrals.

A nonlinear integral may be viewed as a nonlinear functional I : M(X) ×F+(X) → [0,∞], where M(X) is the set of all nonadditive measures µ : A →[0,∞] and F+(X) is the set of all A-measurable functions f : X → [0,∞]. So weformulate our general type of convergence theorem for such a functional. In par-ticular we announce that the monotone, bounded, and dominated convergencetheorems for nonlinear integrals follow from our convergence theorems for func-tionals regardless of the type of nonlinear integrals. A key tool is a perturbation

53

of functional that manages not only the monotonicity of a functional I but alsothe small change of the value I(µ, f) arising as a result of adding small amountsto a measure µ and a function f in the domain of I.

2 One of main results

To state mathematically one of our main results we first collect some necessarydefinitions. A nonadditive measure is a set function µ : A → [0,∞] such thatµ(∅) = 0 and µ(A) ≤ µ(B) whenever A,B ∈ A and A ⊂ B. It is called finite ifµ(X) <∞.

Definition 1. A functional I : M(X)×F+(X) → [0,∞] is called an integral ifit satisfies the following conditions:

(i) I(µ, 0) = I(0, f) = 0 for every µ ∈ M(X) and f ∈ F+(X).(ii) I is jointly monotone, that is, I(µ, f) ≤ I(ν, g) for every µ, ν ∈ M(X) with

µ ≤ ν and f, g ∈ F+(X) with f ≤ g.

Definition 2. Let µ, ν : A → [0,∞] be set functions and f, g ∈ F+(X). Thepair (µ, f) is called dominated by (ν, g) and written (µ, f) ≺ (ν, g) if µ({f ≥t}) ≤ ν({g ≥ t}) for every t ∈ R.

Let Φ denote the set of all functions φ : [0,∞) → [0,∞) satisfying φ(0) =limt→+0 φ(t) = 0. A function belonging to Φ is called a control function.

Definition 3. An integral functional I : M(X)×F+(X) → [0,∞] is called per-turbative if, for each p, q > 0, there are control functions φp,q, ψp,q ∈ Φ satisfyingthe following perturbation: for any µ ∈ M(X), f, g ∈ F+(X), ε ≥ 0, and δ ≥ 0,it holds that

I(µ, f) ≤ I(µ, g) + φp,q(δ) + ψp,q(ε)

whenever ∥f∥µ < p, ∥g∥µ < p, µ(X) < q, and (µ, f) ≺ (µ+ δ, g+ε), where ∥f∥µis the µ-essential supremum of f .

The following is one of our main results and gives a unified formulation of thebounded convergence theorem for linear and nonlinear integrals.

Theorem 1. Let I : M(X) × F+(X) → [0,∞] be an integral functional. Letµ ∈ M(X) be finite. Assume that µ is autocontinuous, that is, µ(A∪Bn) → µ(A)and µ(A \Bn) → µ(A) whenever A,Bn ∈ A (n = 1, 2, . . . ) and µ(Bn) → 0. If Iis perturbative, then the bounded convergence theorem holds for I with respect toµ, that is, for any uniformly µ-essentially bounded sequence {fn}n∈N ⊂ F+(X),if fn converges in µ-measure to a function f ∈ F+(X), then f is µ-essentiallybounded and I(µ, fn) → I(µ, f).

The Lebesgue, the Choquet, the Sipos, the Sugeno, and the Shilkret integralsare all perturbative. Other types of convergence theorems will be also announcedduring the talk.

54

Aggregation on some posets of aggregationfunctions

Anna Kolesarova1 and Radko Mesiar2

1Slovak University of Technology, Fac. of Chemical and Food Technology,Radlinskeho 9, 812 37 Bratislava, Slovakia

2Slovak University of Technology, Fac. of Civil Engineering,Radlinskeho 11, 810 05 Bratislava, Slovakia{anna.kolesarova;radko.mesiar}@stuba.sk

For a fixed natural number n ≥ 2, the set An of all n–ary aggregation func-tions A : [0, 1]n → [0, 1], equipped with the standard partial ordering ≤ of n–aryreal functions, is a complete lattice. More details on aggregation functions canbe found in [1, 2, 6, 7].

There are several subposets of An which form a bounded poset or even abounded (complete) lattice. The aim of this contribution is to discuss aggregationon some of these posets. We recall several aggregation functions on the latticeSn of all semicopulas [4, 5], on the lattice Q2 of all binary quasi-copulas [10],and also on the poset C2 of all binary copulas [10]. As particular examples recallaggregation of quasi-copulas based on aggregation of the corresponding adjointoperators [5] and the Darsow product of copulas [3].

Another distinguished lattice in An is formed by OWA operators [11] (forn = 2 we even get a chain). Aggregation of n–ary OWA operators is discussedfrom several points of view. In some cases the orness/andness parameters ofOWA operators are also considered. Some details can be found in [9]. A similardiscussion of aggregation of the Choquet integrals is also provided. We alsointroduce examples with rather poor results. This is, e.g., the case of the posetof triangular norms (or triangular conorms) [8], or the anti-chain of all weightedarithmetic means. Moreover, some open problems are outlined.

Acknowledgment The authors kindly acknowledge the support of the projectof Science and Technology Assistance Agency under the contract No. APVV–14–0013.

References

1. G. Beliakov, A. Pradera, and T. Calvo, Aggregation Functions: A Guide for Prac-titioners. Springer, 2007.

2. G. Beliakov, H. Bustince, and T. Calvo, A Practical Guide to Averaging Functions.Studies in Fuzziness and Soft Computing 329, Springer, 2016.

3. W. F. Darsow, B. Nguyen, E. T. Olsen, Copulas and Markov processes. Illinois J.Math. 36 (1992) 600-642.

4. F. Durante, C. Sempi, Semicopulae. Kybernetika 41 (2005) 315–328.

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2 A. Kolesarova, R. Mesiar

5. F. Durante, E.P. Klement, R. Mesiar, and C. Sempi: Conjunctors and their residualimplicators: Characterizations and construction methods. Mediterranean Journalof Mathematics 4(3) (2007) 343–356.

6. M. Gagolewski, Data Fusion: Theory, Methods, and Applications. Institute of Com-puter Science, Polish Academy of Sciences, 2015.

7. M. Grabisch, J.-L. Marichal, R. Mesiar, and E. Pap, Aggregation Functions. Cam-bridge University Press, Cambridge, 2009.

8. E. P. Klement, R. Mesiar, E. Pap, Triangular Norms. Kluwer Academic Publishers,Dordrecht, 2000.

9. R. Mesiar, L. Sipeky, P. Gupta, Jin Le Sheng, Aggregation of OWA operators.Working paper, 2016.

10. R. B. Nelsen, An Introduction to Copulas. Second ed. Lecture Notes in Statistics139, Springer, New York, 2006.

11. R. Yager, On ordered weighted averaging aggregation operators in multi-criteriadecision making. IEEE Transactions on Systems, Man and Cybernetics 18 (1988)183–190.

56

Some properties of (C, I)-equivalences

Anna Krol

Interdisciplinary Centre for Computational Modelling,Faculty of Mathematics and Natural Sciences,

University of Rzeszow, Pigonia 1, 35-310 Rzeszow, Polandannakrol@ur.edu.pl

This contribution deals with the notion of a fuzzy equivalence, as one of fuzzyconnectives, whose definition depends on a fuzzy conjunction and implication.More precisely, let C, I be a fuzzy conjunction and implication, respectively.The function E : [0, 1]2 → [0, 1] given by the formula

EC,I(x, y) = C(I(x, y), I(y, x)), x, y ∈ [0, 1]

will be called (C, I)-equivalence. Some properties of (C, I)-equivalences accordingto axioms of other notions of a fuzzy equivalence are presented. Some additionalproperties of (C, I)-equivalences, taking into consideration relevant properties ofgenerators C and I are examined. Moreover, preservation of properties of (C, I)-equivalences in an aggregation process is indicated. Examination of preservationof axioms and properties of fuzzy connectives finds its applications e.g. in decisionmaking, approximate reasoning and fuzzy control.

References

1. Baczynski, M., Jayaram B.: Fuzzy implications. Springer, Berlin, 2008.2. Bentkowska, U., Krol, A.: Generating of fuzzy α–C–equivalences by aggregation

process (submitted).3. Calvo T., Kolesarova, A., Komornikova, M., Mesiar, R.: Aggregation operators:

Properties, classes and construction methods. In: Aggregation Operators, T. Calvoet al. (Eds.), Physica-Verlag, Heildelberg, pp 3-104 (2002).

4. Drewniak, J., Krol A.: A survey of weak connectives and the preservation of theirproperties by aggregations. Fuzzy Sets and Systems 161, 202–215 (2010).

5. Fodor, J. C., Roubens M.: Fuzzy preference modelling and multicriteria decisionsupport. Kluwer, Dordrecht, 1994.

6. Klement, E. P., Mesiar R., Pap E.: Triangular norms. Kluwer Academic Publishers,Dordrecht, 2000.

7. Pradera A., Beliakov, G., Bustince, H., De Baets B.: A review of the relationshipsbetween implication, negation and aggregation functions from the point of view ofmaterial implication. Information Sciences 329, 357–380 (2016).

Acknowledgement This work was partially supported by the Centre for Innovationand Transfer of Natural Sciences and Engineering Knowledge in Rzeszow, throughProject Number RPPK.01.03.00-18-001/10.

57

Extremal weighted aggregation

Radko Mesiar and Andrea Stupnanova

Slovak University of Technology, Faculty of Civil EngineeringRadlinskeho 11,

810 05 Bratislava, Slovak Republic{radko.mesiar,andrea.stupnanova}@stuba.sk

Weighted aggregation functions allow to introduce weights or importances ofsingle inputs into the global aggregation. For more details see Chapter 6.4. ofmonograph [1]. In the standard introduction of weights we consider two n- tuples:the weighting vector w = (w1, . . . , wn) with some constraints (w ∈ [0, 1]

nand

n∑i=1

wi = 1, orn∨i=1

wi = 1, etc.), and the input vector x = (x1, . . . , xn) (mostly

x ∈ [0, 1]n

or x ∈ Rn). The order of weights and inputs is usually fixed, see,for example weighted (quasi-)arithmetic means. In some cases, only the order ofweights is fixed, see OWA operators, for example.The aim of this contribution is to discuss the case when neither the orderingof weights nor of the inputs is fixed, i.e., when any weight wi can be assignedto any input xj . Formally, we have (n!)2 different situations to be discussed,characterized by

(wσ,xτ ) = ((wσ(1), xτ(1)), . . . , (wσ(n), xτ(n))),

where σ, τ : {1, . . . , n} → {1, . . . , n} are permutations.We denote by Pn the set of all such permutations. For a considered weighted ag-gregation function A : [0, 1]

n× [0, 1]n → [0, 1], we introduce two new aggregation

functions A∗, A∗ : [0, 1]n × [0, 1]

n → [0, 1] given by

A∗(w,x) = max{A(wσ,xτ )|σ, τ ∈ Pn}and (1)

A∗(w,x) = min{A(wσ,xτ )|σ, τ ∈ Pn}

Clearly, these extremal weighted aggregation functions satisfy

A∗(w,x) ≤ A(wσ,xτ ) ≤ A∗(w,x)

for all w,x ∈ [0, 1]n

and σ, τ ∈ Pn.We discuss and study some particular cases. So, for example, consider the

weighted arithmetic mean W : [0, 1]n × [0, 1]

n → [0, 1] given by

W (w,x) =

n∑i=1

wixi (where

n∑i=1

wi = 1).

ThenW ∗(w,x) = W (wα,xβ)

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2 Extremal weighted aggregation

for any permutations α, β ∈ Pn such that the n-tuples wα and xβ are comono-tone. In particular,

W ∗(w,x) = W (w∗x,x),

where w∗x = wγ is comonotone with x, γ ∈ Pn. Similarly,

W∗(w,x) = W (wx∗,x),

where wx∗ = wγ−1 is countermonotone with x.For OWA operators we have

OWA∗(w,x) = OWA(w∗,x),

where w∗ = wδ, wδ(1) ≥ · · · ≥ wδ(n), and

OWA∗(w,x) = OWA(w∗,x),

where w∗ = wδ = (wδ(n), . . . , wδ(1)). Obviously, for any weighted aggregationfunction A, both A∗ and A∗ are symmetric aggregation functions. More, theidempotency of A, A∗ and A∗ either holds for all three aggregation functions,or for none of them.

We include also a discussion on extremal capacity - based integrals, where in-stead of an n-dimensional weighting vector w ∈ [0, 1]

n, a capacity m : 2{1,...,n} →

[0, 1] is considered [2],[3],[4].

Acknowledgments. The support of the grants APVV-14-0013 and VEGA1/0682/16 is kindly announced.

References

1. Grabisch, M., Marichal, J.L., Mesiar, R., Pap, E.: Aggregation Functions. Cam-bridge University Press, Cambridge (2009)

2. Klement, E.P., Mesiar, R., Pap, E.: A universal integral as common frame forChoquet and Sugeno integral. IEEE Transactions on Fuzzy Systems 18, 178–187(2010)

3. Lehrer, E: A new integral for capacities. Economic Theory 39, 157–176 (2009)4. Mesiar, R., Stupnanova, A.: Decomposition integrals. International Journal of Ap-

proximate Reasoning 54, 1252–1259 (2013)

59

Monotonicity:A cornerstone of social choice theory

Raul Perez-Fernandez and Bernard De Baets

KERMIT, Department of Mathematical Modelling, Statistics and Bioinformatics,Ghent University, Coupure links 653, B-9000 Gent, Belgium{raul.perezfernandez,bernard.debaets}@ugent.be

Abstract. Electing a winner based on the preferences expressed by sev-eral voters over a set of candidates has been a relevant matter of studyfor centuries. Although some examples of voting procedures have beenproved to be used already in the Ancient Greece [19], it is in the eigh-teenth century, with the works of Rousseau [17], Borda [2] and Con-dorcet [4], that social choice theory became a theoretical framework call-ing the attention of the scientific community.

In particular, we discuss here the sub-problem of ranking candidatesaccording to the preferences of the voters, where each voter expresseshis/her preferences in the form of a profile of rankings (list of rankings)on the set of candidates. Unfortunately, for some profiles, depending onthe choice of ranking rule (function selecting a winning ranking givenany profile of rankings), one can obtain a different ranking as output.Actually, Arrow’s Theorem [1] states that there is no ranking rule si-multaneously satisfying a set of properties (non-dictatorship, unanim-ity and independence of irrelevant alternatives) that can be considerednatural and desirable. This fact leads to a vast literature where manydifferent ranking rules are proposed: plurality [19], majority [8,17], theBorda count [2], veto [18], Condorcet’s least reversals [4,5,10] or Ke-meny’s method [6], just mentioning some of the most relevant ones.

Several authors such as Nitzan [11], Lerer and Nitzan [7], Campbell andNitzan [3], Nurmi [12], Meskanen and Nurmi [9,10] and Perez-Fernandezet al. [15] have called attention to the fact that not only the choice ofa ranking rule is necessary but also agreeing on what we understandby consensus state (or simply consensus). Most ranking rules have beenproved [3,10] to be characterized by a consensus state and a distancefunction1. In case the given profile of rankings is in the chosen consensusstate it is selected as the winning ranking; otherwise the search for theclosest profile of rankings in the chosen consensus state is addressed.

A trivial consensus state is unanimity [6], where each voter has the exactsame preferences on the set of candidates. Another slightly more involvedone is the existence of a weak Condorcet ranking [4], which is a rankingwhere every candidate is preferred by at least half of the voters to all thecandidates ranked after him/her. Another example of consensus state is

1 In [15], we stated that distance functions are actually too strict and proposed toconsider monometrics instead.

60

based on the notion of monotonicity of the votrix2 [13,16]. For a rankinga � b � c, monotonicity of the votrix means that the number of voterspreferring a to c should not be less than both the number of voterspreferring a to b and the number of voters preferring b to c. Relativepositions between candidates are not gathered by the votrix and, in orderto take this hitherto unexploited information into account, we proposedin [13] a new representation of votes, the votex, that is a natural extensionof the votrix. After all, when a voter is providing a ranking a � b � c,he is actually declaring that he supports a over c stronger than botha over b and b over c, something that is not gathered by the votrix.Monotonicity of the votrix and monotonicity of the votex lead to twoconsensus states that are located in between unanimity (which is a toorestrictive consensus state) and the presence of a weak Condorcet ranking(which is a too weak consensus state usually leading to the well-known‘voting paradox’).

Anyhow, monotonicity can also be considered with respect to other rep-resentations of votes, such us the scorix [14] or the (contracted) profile.All these monotonicity-based consensus states lead to different types ofcompromises and the relation between all of them will be the main focusof the presentation.

References

1. K. J. Arrow, A difficulty in the concept of social welfare, The Journal of PoliticalEconomy 58 (4) (1950) 328–346.

2. J. C. Borda, Memoire sur les Elections au Scrutin, Histoire de l’Academie Royaledes Sciences, Paris, 1781.

3. D. E. Campbell, S. I. Nitzan, Social compromise and social metrics, Social Choiceand Welfare 3 (1986) 1–16.

4. M. Condorcet, Essai sur l’Application de l’Analyse a la Probabilite des DecisionsRendues a la Pluralite des Voix, De l’Imprimerie Royale, Paris, 1785.

5. C. L. Dodgson, A Method of Taking Votes on More than Two Issues, ClarendonPress, Oxford, 1876.

6. J. G. Kemeny, Mathematics without numbers, Daedalus 88 (4) (1959) 577–591.

7. E. Lerer, S. Nitzan, Some general results on the metric rationalization for socialdecision rules, Journal of Economic Theory 37 (1985) 191–201.

8. K. O. May, A set of independent necessary and sufficient conditions for simplemajority decision, Econometrica 20 (1952) 680–684.

9. T. Meskanen, H. Nurmi, Distance from consensus: A theme and variations, in:B. Simeone, F. Pukelsheim (eds.), Mathematics and Democracy. Recent Advancesin Voting Systems and Collective Choice, Springer, Berlin, 2006, pp. 117–132.

10. T. Meskanen, H. Nurmi, Closeness counts in social choice, in: M. Braham, F. Steffen(eds.), Power, Freedom, and Voting, Springer, Heidelberg, 2008, pp. 289–306.

11. S. Nitzan, Some measures of closeness to unanimity and their implications, Theoryand Decision 13 (1981) 129–138.

2 A votrix is a reciprocal matrix that is the voting matrix [4,20] (also known as matrixof pairwise comparisons) of at least one profile of rankings.

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12. H. Nurmi, Settings of consensual processes: Candidates, verdicts, policies, in:E. Herrera-Viedma, J. L. Garcıa-Lapresta, J. Kacprzyk, M. Fedrizzi, H. Nurmi,S. Zadrozny (eds.), Consensual Processes, Springer, Berlin, 2011, pp. 159–177.

13. R. Perez-Fernandez, M. Rademaker, P. Alonso, I. Dıaz, S. Montes, B. De Baets,Representations of votes facilitating monotonicity-based ranking rules: From votrixto votex, International Journal of Approximate Reasoning 73 (2016) 87–107.

14. R. Perez-Fernandez, M. Rademaker, B. De Baets, The scorix: A popular represen-tation of votes revisited, International Journal of Approximate Reasoning, submit-ted.

15. R. Perez-Fernandez, M. Rademaker, B. De Baets, Monometrics and their role inthe rationalisation of ranking rules, Information Fusion 34 (2017) 16–27.

16. M. Rademaker, B. De Baets, A ranking procedure based on a natural monotonicityconstraint, Information Fusion 17 (2014) 74–82.

17. J. J. Rousseau, Du Contrat Social, Marc Michel Rey, Amsterdam, 1762.18. D. G. Saari, Basic Geometry of Voting, Springer-Verlag, Heildelberg, 1995.19. E. S. Staveley, Greek and Roman Voting and Elections, Cornell University Press,

New York, 1972.20. H. P. Young, Condorcet’s theory of voting, American Political Science Review

82 (4) (1988) 1231–1244.

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Dépôt légal: Juillet 2016Imprimé au Grand-Duché de Luxembourg

ILIAS

ISBN 978-99959-0-241-4