research article regularity in terms of hyperidealsand studied the notion of a partial -ary...

Hindawi Publishing CorporationChinese Journal of MathematicsVolume 2013, Article ID 167037, 4 pageshttp://dx.doi.org/10.1155/2013/167037

Research ArticleRegularity in terms of Hyperideals

Kostaq Hila,1 Bijan Davvaz,2 Krisanthi Naka,1 and Jani Dine1

1 Department of Mathematics and Computer Science, Faculty of Natural Sciences, University of Gjirokastra,Gjirokastra 6001, Albania

2Department of Mathematics, Yazd University, Yazd, Iran

Correspondence should be addressed to Bijan Davvaz; [email protected]

Received 9 August 2013; Accepted 9 October 2013

Academic Editors: L. Denis and G. Toth

Copyright © 2013 Kostaq Hila et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper deals with the algebraic hypersystems.The notion of regularity of different type of algebraic systems has been introducedand characterized by different authors such as Iseki, Kovacs, and Lajos. We generalize this notion to algebraic hypersystems givinga unified generalization of the characterizations of Kovacs, Iseki, and Lajos. We generalize also the concept of ideal introducingthe notion of 𝑗-hyperideal and hyperideal of an algebraic hypersystem. It turns out that the description of regularity in terms ofhyperideals is intrinsic to associative hyperoperations in general. The main theorem generalizes to algebraic hypersystems someresults on regular semigroups and regular rings and expresses a necessary and sufficient condition bymeans of principal hyperideals.Furthermore, two more theorems are obtained: one is concerned with a necessary and sufficient condition for an associative,commutative algebraic hypersystem to be regular and the other is concerned with nilpotent elements in the algebraic hypersystem.

1. Introduction

Algebraic structures play a prominent role in mathematicswith wide ranging applications in many disciplines such astheoretical physics, computer sciences, control engineering,information sciences, and coding theory.

Hyperstructure theory was introduced in 1934, whenMarty [1] defined hypergroups based on the notion of hyper-operation, began to analyze their properties, and appliedthem to groups. In the following decades and nowadays, anumber of different hyperstructures are widely studied fromthe theoretical point of view and for their applications tomany subjects of pure and applied mathematics by manymathematicians. In a classical algebraic structure, the com-position of two elements is an element, while in an algebraichyperstructure, the composition of two elements is a set.Several books have been written on hyperstructure theory;see [2–5]. A recent book on hyperstructures [3] points outtheir applications in rough set theory, cryptography, codes,automata, probability, geometry, lattices, binary relations,graphs, and hypergraphs. Another book [4] is devoted espe-cially to the study of hyperring theory. Several kinds of hyper-rings are introduced and analyzed. The volume ends with an

outline of applications in chemistry and physics, analyzingseveral special kinds of hyperstructures: 𝑒-hyperstructuresand transposition hypergroups. The theory of suitable modi-fied hyperstructures can serve as a mathematical backgroundin the field of quantum communication systems.𝑛-ary generalizations of algebraic structures are the most

natural way for further development and deeper understand-ing of their fundamental properties [6, 7]. In [8], Davvaz andVougiouklis introduced the concept of 𝑛-ary hypergroups asa generalization of hypergroups in the sense of Marty. Also,we can consider 𝑛-ary hypergroups as a nice generalization of𝑛-ary groups. Leoreanu-Fotea and Davvaz in [9] introducedand studied the notion of a partial 𝑛-ary hypergroupoid,associated with a binary relation. Some important results,concerning Rosenberg partial hypergroupoids, induced byrelations, are generalized to the case of 𝑛-ary hypergroupoids.Davvaz et al. in [10, 11] considered a class of algebraic hyper-systems which represent a generalization of semigroups,hypersemigroups, and 𝑛-ary semigroups. In this paper wedeal with the algebraic hypersystems.The notion of regularityof different type of algebraic systems has been introduced andcharacterized by different authors such as Iseki [12], Kovacs[13], and Lajos [14]. We generalize this notion to algebraic

2 Chinese Journal of Mathematics

hypersystems giving a unified generalization of the charac-terizations of Kovacs, Iseki, and Lajos. We generalize alsothe concept of ideal introducing the notion of 𝑗-hyperidealand hyperideal of an algebraic hypersystem. It turns outthat the description of regularity in terms of hyperidealsis intrinsic to associative hyperoperations in general. Themain theorem generalizes to algebraic hypersystems someresults on regular semigroups and regular rings and expressesa necessary and sufficient condition by means of principalhyperideals. Furthermore, two more theorems are obtained:one is concerned with a necessary and sufficient conditionfor an associative, commutative algebraic hypersystem to beregular; another is concerned with nilpotent elements in thealgebraic hypersystem.

2. Algebraic Hypersystems and𝑚-Ary Hyperstructures

In this section we recall some known notions on what ismeant by an algebraic hypersystem and 𝑚-ary hyperstruc-ture.

Let𝐻 be a nonempty set and 𝑓 a mapping 𝑓 : 𝐻 ×𝐻 →

P∗(𝐻), whereP∗(𝐻) denotes the set of all nonempty subsetsof 𝐻. Then 𝑓 is called a binary (algebraic) hyperoperationon 𝐻. In general, a mapping 𝑓 : 𝐻 × 𝐻 × ⋅ ⋅ ⋅ × 𝐻 →

P∗(𝐻), where 𝐻 appears 𝑚 times, is called an 𝑚-ary(algebraic) hyperoperation, and 𝑚 is called the arity of thishyperoperation. An algebraic system (𝐻, 𝑓), where 𝑓 is an𝑚-ary hyperoperation defined on 𝐻, is called an 𝑚-aryhypergroupoid or an𝑚-ary hypersystem. Since we identify theset {𝑥} with the element 𝑥, any𝑚-ary (binary) groupoid is an𝑚-ary (binary) hypergroupoid.

Let 𝑓 an𝑚-ary hyperoperation on𝐻 and𝐴1, 𝐴2, . . . , 𝐴

𝑚

nonempty subsets of𝐻. We define

𝑓 (𝐴1, 𝐴2, . . . , 𝐴

𝑚)

= {𝑓 (𝑥1, 𝑥2, . . . , 𝑥

𝑚) | 𝑥𝑖∈ 𝐴𝑖, 𝑖 = 1, 2, . . . , 𝑚} .

(1)

We will use the following abbreviated notation: thesequence 𝑥

𝑖, 𝑥𝑖+1, . . . , 𝑥

𝑗will be denoted by 𝑥𝑗

𝑖. For 𝑗 < 𝑖, 𝑥𝑗

𝑖

is the empty symbol. In this convention,

𝑓 (𝑥1, . . . , 𝑥

𝑖, 𝑦𝑖+1, . . . , 𝑦

𝑗, 𝑧𝑗+1, . . . , 𝑧

𝑚) (2)

will be written as 𝑓(𝑥𝑖1, 𝑦𝑗

𝑖+1, 𝑧𝑚𝑗+1). In the case when 𝑦

𝑖+1=

⋅ ⋅ ⋅ = 𝑦𝑗= 𝑦, the last expression will be written in the form

𝑓(𝑥𝑖1,(𝑗−𝑖)

𝑦 , 𝑧𝑚𝑗+1).

Similarly, for subsets 𝐴1, 𝐴2, . . . , 𝐴

𝑚of𝐻 we define

𝑓 (𝐴𝑚

1) = 𝑓 (𝐴

1, 𝐴2, . . . , 𝐴

𝑚)

= {𝑓 (𝑥𝑚

1) | 𝑥𝑖∈ 𝐴𝑖, 𝑖 = 1, . . . , 𝑚} .

(3)

An𝑚-ary hyperoperation 𝑓 is called (𝑖, 𝑗)-associative if

𝑓 (𝑥𝑖−1

1, 𝑓 (𝑥𝑚+𝑖−1

𝑖) , 𝑥2𝑚−1

𝑚+𝑖) = 𝑓 (𝑥

𝑗−1

1, 𝑓 (𝑥𝑚+𝑗−1

𝑗) , 𝑥2𝑚−1

𝑚+𝑗)

(4)

holds for fixed 1 ≤ 𝑖 < 𝑗 ≤ 𝑚 and all 𝑥1, 𝑥2, . . . , 𝑥

2𝑚−1∈ 𝐻.

Note that (𝑖, 𝑘)-associativity follows from (𝑖, 𝑗)- and (𝑗, 𝑘)-associativity.

If the above condition is satisfied for all 𝑖, 𝑗 ∈ {1, 2, . . . , 𝑚},then we say that 𝑓 is associative.

The 𝑚-ary hyperoperation 𝑓 is called commutative ifand only if for all 𝑥

1, . . . , 𝑥

𝑚∈ 𝐻 and for all 𝜎 ∈ S

𝑚,

𝑓(𝑥1, 𝑥2, . . . , 𝑥

𝑚) = 𝑓(𝑥

𝜎(1), 𝑥𝜎(2), . . . , 𝑥

𝜎(𝑚)).

By an algebraic hypersystem (𝐻, 𝑓1, 𝑓2, . . . , 𝑓

𝑚) or simply

𝐻 is meant a set 𝐻 closed under a collection of 𝑚𝑖-ary

hyperoperation 𝑓𝑖and often also satisfying a fixed set of laws,

for instance, the associative law.

3. Regular Algebraic Hypersystems

Let𝐻be an algebraic hypersystem.𝐻 is said to be regularwithrespect to the hyperoperation 𝑓 if and only if for each 𝑎 ∈ 𝐻there exist𝑥

2, 𝑥3, . . . , 𝑥

𝑚; 𝑦1, 𝑦3, . . . , 𝑦

𝑚; . . . ; 𝑧

1, 𝑧2, . . . , 𝑧

𝑚−1∈

𝐻 such that

𝑎 ∈ 𝑓 (𝑓 (𝑎, 𝑥2, . . . , 𝑥

𝑚) , 𝑓 (𝑦

1, 𝑎, 𝑦3, . . . , 𝑦

𝑚) , . . . ,

𝑓 (𝑧1, 𝑧2, . . . , 𝑧

𝑚−1, 𝑎)) .

(5)

A subset 𝑆 of𝐻 constitutes a subhypersystem if and only if𝑆 is closed under the same hyperoperations and satisfies thesame fixed laws in𝐻.

Let 𝐻 be an algebraic hypersystem. A 𝑗-hyperideal 𝑗 =1, 2, . . . , 𝑚 relative to the 𝑚-ary hyperoperation is defined tobe a subhypersystem 𝐼

𝑗such that, for any 𝑥

1, 𝑥2, . . . , 𝑥

𝑚∈ 𝐻,

if 𝑥𝑗∈ 𝐼𝑗, then 𝑓(𝑥

1, 𝑥2, . . . , 𝑥

𝑚) ⊆ 𝐼

𝑗. The 𝑗-hyperideal

relative to 𝑓 generated by an element 𝑎 ∈ 𝐻 (usually called aprincipal 𝑗-hyperideal) is denoted by

(𝑎)𝑗= 𝑓(𝐻,𝐻, . . . ,

𝑗

𝑎, . . . , 𝐻) ∪ {𝑎} . (6)

A subhypersystem 𝐼 which is a 𝑗-hyperideal for each 𝑗 =1, . . . , 𝑚 is simply called a hyperideal.

Theorem 1. Let 𝐻 be an algebraic hypersystem which isassociative relative to an 𝑚-ary hyperoperation 𝑓. Then thefollowing conditions are equivalent.

(1) 𝐻 is regular relative to the hyperoperation 𝑓.

(2) 𝑓(𝐼1, 𝐼2, . . . , 𝐼

𝑚) = ⋂

𝑚

𝑗=1𝐼𝑗for any set of 𝑗-hyperideals

𝐼𝑗relative to the hyperoperation.

(3) 𝑓((𝑎1)1, (𝑎2)2, . . . , (𝑎

𝑚)𝑚) = ⋂

𝑚

𝑗=1(𝑎𝑗)𝑗for any set of

elements 𝑎1, 𝑎2, . . . , 𝑎

𝑚∈ 𝐻.

(4) 𝑓((𝑎)1, (𝑎)2, . . . , (𝑎)

𝑚) = ⋂

𝑚

𝑗=1(𝑎)𝑗for each element

𝑎 ∈ 𝐻.

Chinese Journal of Mathematics 3

Proof. “(1) ⇒ (2).” Let 𝐻 be regular relative to the 𝑚-aryhyperoperation 𝑓 and let 𝑎 ∈ ⋂𝑚

𝑗=1𝐼𝑗for any set of 𝑚 𝑗-

hyperideals 𝐼𝑗relative to the hyperoperation.Thenby regular-

ity there exists 𝑥2, . . . , 𝑥

𝑚; 𝑦1, 𝑦3, . . . , 𝑦

𝑚, . . . ; 𝑧

1, . . . , 𝑧

𝑚−1∈

𝐻 such that

𝑎 ∈ 𝑓 (𝑓 (𝑎, 𝑥2, . . . , 𝑥

𝑚) , 𝑓 (𝑦

1, 𝑎, . . . , 𝑦

𝑚) , . . . ,

𝑓 (𝑧1, . . . , 𝑧

𝑚−1, 𝑎)) .

(7)

With 𝐼𝑗being a 𝑗-hyperideal for each 𝑗 = 1, . . . , 𝑚, we

thus obtain 𝑓(𝑎, 𝑥2, . . . , 𝑥

𝑚) ⊆ 𝐼1, 𝑓(𝑦1, 𝑎, . . . , 𝑦

𝑚) ⊆ 𝐼2, . . .,

and 𝑓(𝑧1, 𝑧2, . . . , 𝑧

𝑚−1, 𝑎) ⊆ 𝐼

𝑚and hence ⋂𝑚

𝑗=1𝐼𝑗⊆

𝑓(𝐼1, 𝐼2, . . . , 𝐼

𝑚).

Conversely, if 𝑎 ∈ 𝑓(𝐼1, 𝐼2, . . . , 𝐼

𝑚), then 𝑎 ∈

𝑓(𝑖1, 𝑖2, . . . , 𝑖

𝑚) for 𝑖

𝑗∈ 𝐼𝑗, 𝑗 = 1, . . . , 𝑚, and therefore 𝑎 ∈ 𝐼

𝑗

for each 𝑗 = 1, . . . , 𝑚. Hence, (2) is proved.“(2) ⇒ (3) ⇒ (4)” are obvious.“(4) ⇒ (1).” Let 𝑓((𝑎)

1, (𝑎)2, . . . , (𝑎)

𝑛) = ⋂

𝑚

𝑗=1(𝑎)𝑗for

each 𝑎 ∈ 𝐻. Since for each 𝑎 ∈ 𝐻, 𝑎 ∈ ⋂𝑚𝑗=1(𝑎)𝑗, then 𝑎 ∈

𝑓(𝑏1, 𝑏2, . . . , 𝑏

𝑚), where either 𝑏

𝑘= 𝑎 or 𝑏

𝑘∈ 𝑓(𝑐1, 𝑐2, . . . , 𝑐

𝑚)

with 𝑐𝑘= 𝑎. Thus, we have in any case the following:

𝑎 ∈ 𝑓 (𝑏1, 𝑏2, . . . , 𝑏

𝑚)

= 𝑓 (𝑓 (𝑎, 𝑥2, . . . , 𝑥

𝑚) , 𝑓 (𝑦

1, 𝑎, . . . , 𝑦

𝑚) , . . . ,

𝑓 (𝑧1, 𝑧2, . . . , 𝑧

𝑚−1, 𝑎))

(8)

for some 𝑥2, . . . , 𝑥

𝑚; 𝑦1, 𝑦3, . . . , 𝑦

𝑚; . . . ; 𝑧

1, . . . 𝑧𝑚−1

∈ 𝐻. Thisshows that 𝐻 is regular with respect to the hyperoperation.

Theorem 2. An algebraic hypersystem 𝐻 which is associativeand commutative relative to an 𝑚-ary hyperoperation 𝑓 isregular with respect to the same hyperoperation if and only ifevery hyperideal 𝐼 of𝐻 is idempotent; that is,𝑓(𝐼, 𝐼, . . . , 𝐼) = 𝐼.

Proof. If 𝐻 is commutative relative to 𝑓, then 𝑓(𝑎,𝐻, . . . , 𝐻) = 𝑓(𝐻, 𝑎, . . . , 𝐻) = ⋅ ⋅ ⋅ = 𝑓(𝐻,𝐻, . . . , 𝑎)and hence every 𝑗-hyperideal is also a 𝑘-hyperideal for all𝑗, 𝑘 = 1, . . . , 𝑚. Hence, by regularity

𝑓 (𝐼, 𝐼, . . . , 𝐼) = 𝐼 ∩ 𝐼 ∩ 𝐼 ∩ ⋅ ⋅ ⋅ ∩ 𝐼 = 𝐼

∀ hyperideal 𝐼 in 𝐻.(9)

Conversely, suppose that every hyperideal in 𝐻 is idem-potent. If 𝐼

1, 𝐼2, . . . , 𝐼

𝑚are hyperideals of 𝐻, then ⋂𝑚

𝑗=1𝐼𝑗is

also an hyperideal and therefore

𝑚

⋂𝑗=1

𝐼𝑗= 𝑓(

𝑚

⋂𝑗=1

𝐼𝑗,𝑚

⋂𝑗=1

𝐼𝑗, . . . ,𝑚

⋂𝑗=1

𝐼𝑗) ⊆ 𝑓 (𝐼

1, 𝐼2, . . . , 𝐼

𝑚) (10)

inasmuch as 𝐼𝑗contains the intersection for each 𝑗. Further-

more, since each 𝐼𝑗, 𝑗 = 1, . . . , 𝑚 is also a 𝑗-hyperideal, then

𝑓(𝐼1, 𝐼2, . . . , 𝐼

𝑚) ⊆ ⋂

𝑚

𝑗=1𝐼𝑗. Hence, the conclusion follows.

By what we mentioned in the beginning of the section,note that in case the𝑚-ary hyperoperation𝑓 is an associative𝑚-ary hyperoperation in𝐻 one may conveniently abbreviate

𝑓 (𝑎, 𝑎, . . . , 𝑎) = 𝑓 (𝑎𝑚

) ,

𝑓 (𝑓 (𝑎𝑚

) , 𝑎, . . . , 𝑎) = 𝑓 (𝑎2𝑚−1

) ,


) , 𝑓 (𝑎𝑚

) , . . . , 𝑎) = 𝑓 (𝑎3𝑚−2

) ,

...


) , 𝑓 (𝑎𝑚

) , . . . , 𝑓 (𝑎𝑚

)) = 𝑓 (𝑎𝑚2

) = 𝑓 (𝑎(𝑚+1)𝑚−𝑚

) .

(11)

Thus, the admissible exponents of compositions of rank atmost 2 are each of the form 𝑘𝑚 − 𝑘 + 1 for some integer.Proceeding inductively, suppose that 𝑘

1𝑚 − 𝑘

1+ 1, 𝑘

2𝑚 −

𝑘2+ 1, . . . , 𝑘

𝑚𝑚 − 𝑘

𝑚+ 1 are previously known admissible

exponents; then the exponent

𝑚

∑𝑖=1

(𝑘𝑖𝑚 − 𝑘

𝑖+ 1) = (

𝑚

∑𝑖=1

𝑘𝑖− 1)𝑚 −

𝑚

∑𝑖=1

𝑘𝑖

(12)

of 𝑓(𝑓(𝑎𝑘1𝑚−𝑘1+1), 𝑓(𝑎𝑘2𝑚−𝑘2+1), . . . , 𝑓(𝑎𝑘𝑚𝑚−𝑘𝑚+1)) is evi-dently also of the same form. Hence, every admissibleexponent of an 𝑚-ary hyperoperation is of the form𝑘𝑚 − 𝑘 + 1.

An element 𝑎 ∈ 𝐻 such that

𝑓 (𝑎, 𝑥1, . . . , 𝑥

𝑚−1) = 𝑓 (𝑥

1, 𝑎, . . . , 𝑥

𝑚−1)

= 𝑓 (𝑥1, . . . , 𝑥

𝑚−1, 𝑎) = {𝑎} ,

(13)

for all 𝑥1, 𝑥2, . . . , 𝑥

𝑚−1∈ 𝐻, is called zero element. The zero

element is denoted by 0. A nilpotent element 𝑎 ∈ 𝐻 is onewhich satisfies 𝑓(𝑎𝑘𝑚−𝑘+1) = {0} for some integer 𝑘 greaterthan 0.

Theorem 3. An algebraic hypersystem 𝐻 which is commuta-tive, associative, regular, and has a 0 with respect to an 𝑚-aryhyperoperation 𝑓 possesses no nilpotent element other than 0.

Proof. For all 0 ̸= 𝑎 ∈ 𝐻, let [𝑎] denote the subhypersystemof 𝐻 generated by 𝑎, which may be inductively defined asfollows:

(1) 𝑎 ∈ [𝑎](2) 𝑓(𝑎𝑚) ⊆ [𝑎](3) whenever 𝑓(𝑎𝑛1), . . . , 𝑓(𝑎𝑛𝑚) ⊆ [𝑎], then also𝑓(𝑎𝑛1+⋅⋅⋅+𝑛𝑚) ⊆ [𝑎].

In order to prove the theorem it suffices to show that 𝑎 ∉ [𝑎].We proceed inductively as follows.

(1) 𝑎 ̸= 0 by assumption.(2) Consider 𝑓(𝑎𝑚) ̸= {0}. For, if 𝑓(𝑎𝑚) = {0}, then

by virtue of the associativity, commutativity, andregularity of the given hyperoperation, there exist

4 Chinese Journal of Mathematics

𝑥1, . . . , 𝑥

𝑚−1∈ 𝐻 such that 𝑎 ∈ 𝑓(𝑎, 𝑥

1, 𝑎, . . . ,

𝑓(. . . , 𝑥𝑚−1, 𝑎)) = 𝑓(𝑓(𝑎, 𝑎, . . . , 𝑎), 𝑥

1, . . . , 𝑥

𝑚−1) =

𝑓(𝑓(𝑎𝑚), 𝑥1, . . . , 𝑥

𝑚−1) = 𝑓(0, 𝑥

1, . . . , 𝑥

𝑚−1) = {0}

contrary to (1).(3) We now show that if 𝑓(𝑎𝑛1), . . . , 𝑓(𝑎𝑛𝑚) are

all nonzero elements of [𝑎] , then 𝑓(𝑓(𝑎𝑛1),𝑓(𝑎𝑛2), . . . , 𝑓(𝑎𝑛𝑚)) = 𝑓(𝑎𝑛1+𝑛2+⋅⋅⋅+𝑛𝑚) ̸= {0}. Suppose𝑓(𝑎𝑛1+𝑛2+⋅⋅⋅+𝑛𝑚) = {0}. Then by the above remark, wehave

𝑛𝑖= 𝑘𝑖𝑚 − 𝑘

𝑖+ 1 for 𝑖 = 1, 2, . . . , 𝑚. (14)

Since 𝑓 is commutative, it may be assumed without loss ofgenerality that 𝑛

1= max

𝑖𝑛𝑖. Then

𝑚𝑛𝑖=𝑚

∑𝑖=1

𝑛𝑖+ (𝑚𝑛

1−𝑚

∑𝑖=1

𝑛𝑖) =𝑚

∑𝑖=1

𝑛𝑖+𝑚

∑𝑖=1

(𝑛𝑖− 𝑛𝑖)

=𝑚

∑𝑖=1

𝑛𝑖+𝑚

∑𝑖=1

[(𝑘1− 𝑘𝑖)𝑚 − (𝑘

1− 𝑘𝑖)]

=𝑚

∑𝑖=1

𝑛𝑖+ (𝑚

∑𝑖=1

(𝑘1− 𝑘𝑖)𝑚

−𝑚

∑𝑖=1

(𝑘1− 𝑘𝑖) − 𝑚 + 2) + (𝑚 − 2)

=𝑚

∑𝑖=1

𝑛𝑖+ {[

𝑚

∑𝑖=1

(𝑘1− 𝑘𝑖) − 1]𝑚

−[𝑚

∑𝑖=1

(𝑘1− 𝑘𝑖) − 2]} + (𝑚 + 2)

=𝑚

∑𝑖=1

𝑛𝑖+ 𝑝 + (𝑚 − 2) ,

(15)

where 𝑝 is an admissible exponent. Hence, by associativity,commutativity, and regularity of the hyperoperation 𝑓, thereexist 𝑥

1, 𝑥2, . . . , 𝑥

𝑚−1∈ 𝐻 such that

{0} ̸= 𝑓 (𝑎𝑛1)

= 𝑓 (𝑓 (𝑎𝑛1) , 𝑥1, 𝑓 (𝑎𝑛1) , . . . , 𝑓 (. . . , 𝑥

𝑚−1, 𝑓 (𝑎𝑛1)))

= 𝑓 (𝑓 (𝑎𝑚𝑛1) , 𝑥1, . . . , 𝑥

𝑚−1)

= 𝑓 (𝑓 (𝑎𝑛1+𝑛2+⋅⋅⋅+𝑛

𝑚+𝑝+(𝑚−2)

) , 𝑥1, . . . , 𝑥

𝑚−1)

= 𝑓 (𝑓 (𝑎𝑛1+𝑛2+⋅⋅⋅+𝑛

𝑚) ,

𝑓 (𝑓 (𝑎𝑝

) , 𝑎, 𝑎, . . . , 𝑎, 𝑥1) , 𝑥2, . . . , 𝑥

𝑚−1)

= 𝑓 (0, 𝑓 (𝑓 (𝑎𝑝

) , 𝑎, 𝑎, . . . , 𝑎, 𝑥1) , 𝑥2, . . . , 𝑥

𝑚−1) = {0} ,

(16)

a contradiction.Thus, every element of [𝑎] is nonzero and theconclusion follows.

Acknowledgments

KostaqHilawishes to express his warmest thanks to ProfessorApostolos Thoma and the Department of Mathematics,University of Ioannina (Greece) for their hospitality duringthe author’s postdoctoral scholarship supported by the StateScholarships Foundation of Hellenic Republic.

References

[1] F. Marty, “Sur une generalization de la notion de group,” inProceedings of the 8th Congress desMathematiciens Scandinaves,pp. 45–49, Stockholm, Sweden, 1934.

[2] P. Corsini, Prolegomena of Hypergroup Theory, Aviani, 2ndedition, 1993.

[3] P. Corsini and V. Leoreanu, Applications of Hyperstructure The-ory, Advances in Mathematics, Kluwer Academic, Dodrecht,The Netherlands, 2003.

[4] B. Davvaz and V. Leoreanu-Fotea,Hyperring Theory and Appli-cations, International Academic Press, Palm Harbor, Fla, USA,2007.

[5] T. Vougiouklis, Hyperstructures and Their Representations,Hadronic Press, Palm Harbor, Fla, USA, 1994.

[6] W. A. Dudek and I. Grozdzinska, “On ideals in regular n-semigroups,”Matematicki Bilten, vol. 3, no. 4, pp. 29–30, 1980.

[7] L. Vainerman and R. Kerner, “On special classes of n-algebras,”Journal of Mathematical Physics, vol. 37, no. 5, pp. 2553–2565,1996.

[8] B. Davvaz and T. Vougiouklis, “N-ary hypergroups,” IranianJournal of Science and Technology, Transaction A, vol. 30, no. 2,pp. 165–174, 2006.

[9] V. Leoreanu-Fotea and B. Davvaz, “n-hypergroups and binaryrelations,” European Journal of Combinatorics, vol. 29, no. 5, pp.1207–1218, 2008.

[10] B. Davvaz, W. A. Dudek, and T. Vougiouklis, “A generalizationof n-ary algebraic systems,” Communications in Algebra, vol. 37,no. 4, pp. 1248–1263, 2009.

[11] B. Davvaz, W. A. Dudek, and S. Mirvakili, “Neutral elements,fundamental relations and n-ary hypersemigroups,” Interna-tional Journal of Algebra and Computation, vol. 19, no. 4, pp.567–583, 2009.

[12] K. Iseki, “A characterization of regular semigroups,” Proceedingsof the Japan Academy, vol. 32, pp. 676–677, 1995.

[13] L. Kovacs, “A note on regular rings,”PublicationesMathematicaeDebrecen, vol. 4, pp. 465–468, 1956.

[14] S. Lajos, “A remark on regular semigroups,” Proceedings of theJapan Academy, vol. 37, pp. 29–30, 1961.

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