# Regularity in terms of Hyperideals

Authors: Kostaq Hila; Bijan Davvaz; Krisanthi Naka; Jani Dine
Journal: Chinese Journal of Mathematics
 (2013)

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## Abstract

This paper deals with the algebraic hypersystems. The notion of regularity of different type of algebraic systems has been introduced and characterized by different authors such as Iseki, Kovacs, and Lajos. We generalize this notion to algebraic hypersystems giving a unified generalization of the characterizations of Kovacs, Iseki, and Lajos. We generalize also the concept of ideal introducing the notion ofj-hyperideal and hyperideal of an algebraic hypersystem. It turns out that the description of regularity in terms of hyperideals is intrinsic to associative hyperoperations in general. The main theorem generalizes to algebraic hypersystems some results on regular semigroups and regular rings and expresses a necessary and sufficient condition by means 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.


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## Body

## 1. Introduction

Algebraic structures play a prominent role in mathematics with wide ranging applications in many disciplines such as theoretical physics, computer sciences, control engineering, information sciences, and coding theory.Hyperstructure theory was introduced in 1934, when Marty [1] defined hypergroups based on the notion of hyperoperation, began to analyze their properties, and applied them to groups. In the following decades and nowadays, a number of different hyperstructures are widely studied from the theoretical point of view and for their applications to many subjects of pure and applied mathematics by many mathematicians. In a classical algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, 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 out their applications in rough set theory, cryptography, codes, automata, probability, geometry, lattices, binary relations, graphs, and hypergraphs. Another book [4] is devoted especially to the study of hyperring theory. Several kinds of hyperrings are introduced and analyzed. The volume ends with an outline of applications in chemistry and physics, analyzing several special kinds of hyperstructures: e-hyperstructures and transposition hypergroups. The theory of suitable modified hyperstructures can serve as a mathematical background in the field of quantum communication systems.n-ary generalizations of algebraic structures are the most natural way for further development and deeper understanding of their fundamental properties [6, 7]. In [8], Davvaz and Vougiouklis introduced the concept of n-ary hypergroups as a generalization of hypergroups in the sense of Marty. Also, we can consider n-ary hypergroups as a nice generalization of n-ary groups. Leoreanu-Fotea and Davvaz in [9] introduced and studied the notion of a partial n-ary hypergroupoid, associated with a binary relation. Some important results, concerning Rosenberg partial hypergroupoids, induced by relations, are generalized to the case of n-ary hypergroupoids. Davvaz et al. in [10, 11] considered a class of algebraic hypersystems which represent a generalization of semigroups, hypersemigroups, and n-ary semigroups. In this paper we deal with the algebraic hypersystems. The notion of regularity of different type of algebraic systems has been introduced and characterized by different authors such as Iseki [12], Kovacs [13], and Lajos [14]. We generalize this notion to algebraic hypersystems giving a unified generalization of the characterizations of Kovacs, Iseki, and Lajos. We generalize also the concept of ideal introducing the notion of j-hyperideal and hyperideal of an algebraic hypersystem. It turns out that the description of regularity in terms of hyperideals is intrinsic to associative hyperoperations in general. The main theorem generalizes to algebraic hypersystems some results on regular semigroups and regular rings and expresses a necessary and sufficient condition by means 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; another is concerned with nilpotent elements in the algebraic hypersystem.


## 2. Algebraic Hypersystems andm-Ary Hyperstructures

In this section we recall some known notions on what is meant by an algebraic hypersystem andm-ary hyperstructure.LetH be a nonempty set andfa mapping f:H×H→𝒫*(H), where 𝒫*(H) denotes the set of all nonempty subsets of H. Thenfis called a binary (algebraic) hyperoperation onH. 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 this hyperoperation. An algebraic system (H,f), wherefis an m-ary hyperoperation defined on H, is called an m-ary hypergroupoid or an m-ary hypersystem. Since we identify the set {x} with the element x, any m-ary (binary) groupoid is an m-ary (binary) hypergroupoid.Letfan m-ary hyperoperation on H and A1,A2,…,Am nonempty subsets of H. We define
(1)f(A1,A2,…,Am)={f(x1,x2,…,xm)∣xi∈Ai,i=1,2,…,m}.We will use the following abbreviated notation: the sequencexi,xi+1,…,xj will be denoted by xij. For j<i,xij is the empty symbol. In this convention,
(2)f(x1,…,xi,yi+1,…,yj,zj+1,…,zm)
will be written as f(x1i,yi+1j,zj+1m). In the case when yi+1=⋯=yj=y, the last expression will be written in the form f(x1i,y(j-i),zj+1m).Similarly, for subsetsA1,A2,…,Am of H we define
(3)f(A1m)=f(A1,A2,…,Am)={f(x1m)∣xi∈Ai,i=1,…,m}.Anm-ary hyperoperation f is called (i,j)-associative if
(4)f(x1i-1,f(xim+i-1),xm+i2m-1)=f(x1j-1,f(xjm+j-1),xm+j2m-1)
holds for fixed 1≤i<j≤m and all x1,x2,…,x2m-1∈H.Note that(i,k)-associativity follows from(i,j)- and(j,k)-associativity.If the above condition is satisfied for alli,j∈{1,2,…,m}, then we say that f is associative.Them-ary hyperoperation f is called commutative if and only if for all x1,…,xm∈H and for all σ∈𝕊m, f(x1,x2,…,xm)=f(xσ(1),xσ(2),…,xσ(m)).By an algebraichypersystem(H,f1,f2,…,fm) or simply H is meant a set H closed under a collection of mi-ary hyperoperation fi and often also satisfying a fixed set of laws, for instance, the associative law.


## 3. Regular Algebraic Hypersystems

LetH be an algebraic hypersystem. H is said to be regular with respect to the hyperoperation f if and only if for each a∈H there exist x2,x3,…,xm;y1,y3,…,ym;…;z1,z2,…,zm-1∈H such that
(5)a∈f(f(a,x2,…,xm),f(y1,a,y3,…,ym),…,llllllllllf(z1,z2,…,zm-1,a)).A subsetS of H constitutes a subhypersystem if and only if S is closed under the same hyperoperations and satisfies the same fixed laws in H.LetH be an algebraic hypersystem. A j-hyperidealj=1,2,…,m relative to the m-ary hyperoperation is defined to be a subhypersystem Ij such that, for any x1,x2,…,xm∈H, if xj∈Ij, then f(x1,x2,…,xm)⊆Ij. The j-hyperideal relative to f generated by an element a∈H (usually called a principal j-hyperideal) is denoted by
(6)(a)j=f(H,H,…,aj,…,H)∪{a}.A subhypersystemI which is a j-hyperideal for each j=1,…,m is simply called a hyperideal.Theorem 1.
LetH be an algebraic hypersystem which is associative relative to an m-ary hyperoperation f. Then the following conditions are equivalent.(1)
H is regular relative to the hyperoperation f.(2)
f(I1,I2,…,Im)=⋂j=1mIj for any set of j-hyperideals Ij relative to the hyperoperation.(3)
f((a1)1,(a2)2,…,(am)m)=⋂j=1m(aj)j for any set of elements a1,a2,…,am∈H.(4)
f((a)1,(a)2,…,(a)m)=⋂j=1m(a)j  for each element a∈H.Proof.
“
(
1
)
⇒
(
2
)
.
” Let H be regular relative to the m-ary hyperoperation f and let a∈⋂j=1mIj for any set of mj-hyperideals Ij relative to the hyperoperation. Then by regularity there exists x2,…,xm;y1,y3,…,ym,…;z1,…,zm-1∈H such that
(7)a∈f(f(a,x2,…,xm),f(y1,a,…,ym),…,llllllllllf(z1,…,zm-1,a)).
With Ij being a j-hyperideal for each j=1,…,m, we thus obtain f(a,x2,…,xm)⊆I1, f(y1,a,…,ym)⊆I2,…, and f(z1,z2,…,zm-1,a)⊆Im and hence ⋂j=1mIj⊆f(I1,I2,…,Im).
Conversely, ifa∈f(I1,I2,…,Im), then a∈f(i1,i2,…,im) for ij∈Ij, j=1,…,m, and therefore a∈Ij for each j=1,…,m. Hence, (2) is proved.
“
(
2
)
⇒
(
3
)
⇒
(
4
)
” are obvious.
“
(
4
)
⇒
(
1
)
.
” Let f((a)1,(a)2,…,(a)n)=⋂j=1m(a)j for each a∈H. Since for each a∈H, a∈⋂j=1m(a)j, then a∈f(b1,b2,…,bm), where eitherbk=a or bk∈f(c1,c2,…,cm) with ck=a. Thus, we have in any case the following:
(8)a∈f(b1,b2,…,bm)=f(f(a,x2,…,xm),f(y1,a,…,ym),…,llllllllnf(z1,z2,…,zm-1,a))
for some x2,…,xm;y1,y3,…,ym;…;z1,…zm-1∈H. This shows that H is regular with respect to the hyperoperation.Theorem 2.
An algebraic hypersystemH which is associative and commutative relative to an m-ary hyperoperation f is regular with respect to the same hyperoperation if and only if every hyperideal I of H is idempotent; that is, f(I,I,…,I)=I.Proof.
IfH 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 all j,k=1,…,m. Hence, by regularity
(9)f(I,I,…,I)=I∩I∩I∩⋯∩I=I∀hyperidealIinH.
Conversely, suppose that every hyperideal inHis idempotent. If I1,I2,…,Im are hyperideals of H, then ⋂j=1mIj is also an hyperideal and therefore
(10)⋂j=1m‍Ij=f(⋂j=1m‍Ij,⋂j=1m‍Ij,…,⋂j=1m‍Ij)⊆f(I1,I2,…,Im)
inasmuch asIjcontains the intersection for eachj. Furthermore, since each Ij, j=1,…,m is also aj-hyperideal, then f(I1,I2,…,Im)⊆⋂j=1mIj. Hence, the conclusion follows.
By what we mentioned in the beginning of the section, note that in case them-ary hyperoperation f is an associative m-ary hyperoperation in H one may conveniently abbreviate
(11)f(a,a,…,a)=f(am),f(f(am),a,…,a)=f(a2m-1),f(f(am),f(am),…,a)=f(a3m-2),⋮f(f(am),f(am),…,f(am))=f(am2)=f(a(m+1)m-m).
Thus, the admissible exponents of compositions of rank at most 2 are each of the form km-k+1 for some integer. Proceeding inductively, suppose that k1m-k1+1,k2m-k2+1,…,kmm-km+1 are previously known admissible exponents; then the exponent
(12)∑i=1m‍(kim-ki+1)=(∑i=1m‍ki-1)m-∑i=1m‍ki
of f(f(ak1m-k1+1),f(ak2m-k2+1),…,f(akmm-km+1)) is evidently also of the same form. Hence, every admissible exponent of anm-ary hyperoperation is of the form km-k+1.
An elementa∈Hsuch that
(13)f(a,x1,…,xm-1)=f(x1,a,…,xm-1)=f(x1,…,xm-1,a)={a},
for all x1,x2,…,xm-1∈H, is called zero element. The zero element is denoted by 0. A nilpotent element a∈H is one which satisfies f(akm-k+1)={0} for some integer k greater than 0.Theorem 3.
An algebraic hypersystemH which is commutative, associative, regular, and has a 0 with respect to an m-ary hyperoperationfpossesses no nilpotent element other than 0.Proof.
For all0≠a∈H, let [a] denote the subhypersystem of H generated by a, which may be inductively defined as follows: (1)
a∈[a](2)
f(am)⊆[a](3)
wheneverf(an1),…,f(anm)⊆[a], then also f(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)
Considerf(am)≠{0}. For, if f(am)={0}, then by virtue of the associativity, commutativity, and regularity of the given hyperoperation, there exist x1,…,xm-1∈H such that a∈f(a,x1,a,…,f(…,xm-1,a))=f(f(a,a,…,a),x1,…,xm-1)=f(f(am),x1,…,xm-1)=f(0,x1,…,xm-1)={0} contrary to (1).(3)
We now show that iff(an1),…,f(anm) are all nonzero elements of [a], then f(f(an1),f(an2),…,f(anm))=f(an1+n2+⋯+nm)≠{0}. Suppose f(an1+n2+⋯+nm)={0}. Then by the above remark, we have
(14)ni=kim-ki+1fori=1,2,…,m.Since f is commutative, it may be assumed without loss of generality that n1=maxini. Then
(15)mni=∑i=1m‍ni+(mn1-∑i=1m‍ni)=∑i=1m‍ni+∑i=1m‍(ni-ni)=∑i=1m‍ni+∑i=1m‍[(k1-ki)m-(k1-ki)]=∑i=1m‍ni+(∑i=1m‍(k1-ki)mmmmmmm-∑i=1m‍(k1-ki)-m+2)+(m-2)=∑i=1m‍ni+{[∑i=1m‍(k1-ki)-1]mmmmmmm-[∑i=1m‍(k1-ki)-2]}+(m+2)=∑i=1m‍ni+p+(m-2),
where p is an admissible exponent. Hence, by associativity, commutativity, and regularity of the hyperoperation f, there exist x1,x2,…,xm-1∈H such that
(16){0}≠f(an1)=f(f(an1),x1,f(an1),…,f(…,xm-1,f(an1)))=f(f(amn1),x1,…,xm-1)=f(f(an1+n2+⋯+nm+p+(m-2)),x1,…,xm-1)=f(f(an1+n2+⋯+nm),nnnnnf(f(ap),a,a,…,a,x1),x2,…,xm-1)=f(0,f(f(ap),a,a,…,a,x1),x2,…,xm-1)={0},
a contradiction. Thus, every element of [a] is nonzero and the conclusion follows.



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*Source: 167037-2013-12-30.xml*