Regularity in terms of Hyperideals

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 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 and the other is concerned with nilpotent elements in the algebraic hypersystem.

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 and 𝑚-Ary Hyperstructures

In this section we recall some known notions on what is meant by an algebraic hypersystem and𝑚-ary hyperstructure.

Let 𝐻 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,…,𝐴_𝑚 nonempty subsets of 𝐻. We define 𝑓𝐴₁,𝐴₂,…,𝐴_𝑚=𝑓𝑥₁,𝑥₂,…,𝑥_𝑚∣𝑥_𝑖∈𝐴_𝑖,𝑖=1,2,…,𝑚. (1)

We will use the following abbreviated notation: the sequence 𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑗 will be denoted by 𝑥^𝑗_𝑖. For 𝑗<𝑖,𝑥^𝑗_𝑖 is the empty symbol. In this convention, 𝑓𝑥₁,…,𝑥_𝑖,𝑦_𝑖+1,…,𝑦_𝑗,𝑧_𝑗+1,…,𝑧_𝑚 (2) will be written as 𝑓(𝑥^𝑖₁,𝑦^𝑗_𝑖+1,𝑧^𝑚_𝑗+1). In the case when 𝑦_𝑖+1=⋯=𝑦_𝑗=𝑦, the last expression will be written in the form 𝑓(𝑥^𝑖₁,𝑦^{(𝑗−𝑖)},𝑧^𝑚_𝑗+1).

Similarly, for subsets 𝐴₁,𝐴₂,…,𝐴_𝑚 of 𝐻 we define 𝑓𝐴^𝑚₁=𝑓𝐴₁,𝐴₂,…,𝐴_𝑚=𝑓𝑥^𝑚₁∣𝑥_𝑖∈𝐴_𝑖,𝑖=1,…,𝑚. (3)

An 𝑚-ary hyperoperation 𝑓 is called (𝑖,𝑗)-associative if 𝑓𝑥₁^𝑖−1,𝑓𝑥_𝑖^{𝑚+𝑖−1},𝑥^2𝑚−1_𝑚+𝑖=𝑓𝑥₁^𝑗−1,𝑓𝑥_𝑗^{𝑚+𝑗−1},𝑥^2𝑚−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 if and only if for all 𝑥₁,…,𝑥_𝑚∈𝐻 and for all 𝜎∈𝕊_𝑚, 𝑓(𝑥₁,𝑥₂,…,𝑥_𝑚)=𝑓(𝑥_𝜎(1),𝑥_𝜎(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 𝑚_𝑖-ary hyperoperation 𝑓_𝑖 and often also satisfying a fixed set of laws, for instance, the associative law.

3. Regular Algebraic Hypersystems

Let 𝐻 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,…,𝑥_𝑚;𝑦₁,𝑦₃,…,𝑦_𝑚;…;𝑧₁,𝑧₂,…,𝑧_𝑚−1∈𝐻 such that 𝑎∈𝑓𝑓𝑎,𝑥₂,…,𝑥_𝑚,𝑓𝑦₁,𝑎,𝑦₃,…,𝑦_𝑚𝑓𝑧,…,₁,𝑧₂,…,𝑧_𝑚−1,𝑎. (5)

A subset 𝑆 of 𝐻 constitutes a subhypersystem if and only if 𝑆 is closed under the same hyperoperations and satisfies the same fixed laws in 𝐻.

Let 𝐻 be an algebraic hypersystem. A 𝑗-hyperideal 𝑗=1,2,…,𝑚 relative to the 𝑚-ary hyperoperation is defined to be a subhypersystem 𝐼_𝑗 such that, for any 𝑥₁,𝑥₂,…,𝑥_𝑚∈𝐻, if 𝑥_𝑗∈𝐼_𝑗, then 𝑓(𝑥₁,𝑥₂,…,𝑥_𝑚)⊆𝐼_𝑗. The 𝑗-hyperideal relative to 𝑓 generated by an element 𝑎∈𝐻 (usually called a principal 𝑗-hyperideal) is denoted by (𝑎)_𝑗=𝑓𝐻,𝐻,…,𝑎^𝑗,…,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 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,…,𝐼_𝑚)=⋂^𝑚_𝑗=1𝐼_𝑗 for any set of 𝑗-hyperideals 𝐼_𝑗 relative to the hyperoperation.(3)𝑓((𝑎₁)₁,(𝑎₂)₂,…,(𝑎_𝑚)_𝑚)=⋂^𝑚_𝑗=1(𝑎_𝑗)_𝑗 for any set of elements 𝑎₁,𝑎₂,…,𝑎_𝑚∈𝐻.(4)𝑓((𝑎)₁,(𝑎)₂,…,(𝑎)_𝑚)=⋂^𝑚_𝑗=1(𝑎)_𝑗  for each element 𝑎∈𝐻.

Proof. “(1)⇒(2).” Let 𝐻 be regular relative to the 𝑚-ary hyperoperation 𝑓 and let 𝑎∈⋂^𝑚_𝑗=1𝐼_𝑗 for any set of 𝑚𝑗-hyperideals 𝐼_𝑗 relative to the hyperoperation. Then by regularity there exists 𝑥₂,…,𝑥_𝑚;𝑦₁,𝑦₃,…,𝑦_𝑚,…;𝑧₁,…,𝑧_𝑚−1∈𝐻 such that 𝑎∈𝑓𝑓𝑎,𝑥₂,…,𝑥_𝑚,𝑓𝑦₁,𝑎,…,𝑦_𝑚𝑓𝑧,…,₁,…,𝑧_𝑚−1,𝑎. (7) With 𝐼_𝑗 being 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_𝑚 and hence ⋂^𝑚_𝑗=1𝐼_𝑗⊆𝑓(𝐼₁,𝐼₂,…,𝐼_𝑚).
Conversely, if 𝑎∈𝑓(𝐼₁,𝐼₂,…,𝐼_𝑚), then 𝑎∈𝑓(𝑖₁,𝑖₂,…,𝑖_𝑚) for 𝑖_𝑗∈𝐼_𝑗, 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)_𝑗 for each 𝑎∈𝐻. Since for each 𝑎∈𝐻, 𝑎∈⋂^𝑚_𝑗=1(𝑎)_𝑗, then 𝑎∈𝑓(𝑏₁,𝑏₂,…,𝑏_𝑚), where either𝑏_𝑘=𝑎 or 𝑏_𝑘∈𝑓(𝑐₁,𝑐₂,…,𝑐_𝑚) with 𝑐_𝑘=𝑎. Thus, we have in any case the following: 𝑎∈𝑓𝑏₁,𝑏₂,…,𝑏_𝑚=𝑓𝑓𝑎,𝑥₂,…,𝑥_𝑚,𝑓𝑦₁,𝑎,…,𝑦_𝑚𝑓𝑧,…,₁,𝑧₂,…,𝑧_𝑚−1,𝑎 (8) for some 𝑥₂,…,𝑥_𝑚;𝑦₁,𝑦₃,…,𝑦_𝑚;…;𝑧₁,…𝑧_𝑚−1∈𝐻. This shows that 𝐻 is regular with respect to the hyperoperation.

Theorem 2. An algebraic hypersystem 𝐻 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. 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 all 𝑗,𝑘=1,…,𝑚. Hence, by regularity 𝑓(𝐼,𝐼,…,𝐼)=𝐼∩𝐼∩𝐼∩⋯∩𝐼=𝐼∀hyperideal𝐼in𝐻. (9)
Conversely, suppose that every hyperideal in𝐻is idempotent. If 𝐼₁,𝐼₂,…,𝐼_𝑚 are hyperideals of 𝐻, then ⋂^𝑚_𝑗=1𝐼_𝑗 is also an hyperideal and therefore ^𝑚_𝑗=1𝐼_𝑗=𝑓^𝑚_𝑗=1𝐼_𝑗,^𝑚_𝑗=1𝐼_𝑗,…,^𝑚_𝑗=1𝐼_𝑗⊆𝑓𝐼₁,𝐼₂,…,𝐼_𝑚 (10) inasmuch as𝐼_𝑗contains the intersection for each𝑗. Furthermore, since each 𝐼_𝑗, 𝑗=1,…,𝑚 is also a𝑗-hyperideal, then 𝑓(𝐼₁,𝐼₂,…,𝐼_𝑚)⊆⋂^𝑚_𝑗=1𝐼_𝑗. Hence, the conclusion follows.
By what we mentioned in the beginning of the section, note that in case the 𝑚-ary hyperoperation 𝑓 is an associative 𝑚-ary hyperoperation in 𝐻 one may conveniently abbreviate 𝑓(𝑎,𝑎,…,𝑎)=𝑓𝑎^𝑚,𝑓𝑓𝑎^𝑚,𝑎,…,𝑎=𝑓𝑎^2𝑚−1,𝑓𝑓𝑎^𝑚,𝑓𝑎^𝑚,…,𝑎=𝑓𝑎^3𝑚−2,⋮𝑓𝑓𝑎^𝑚,𝑓𝑎^𝑚,…,𝑓𝑎^𝑚=𝑓𝑎^𝑚2=𝑓𝑎^{(𝑚+1)𝑚−𝑚}. (11) 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,…,𝑘_𝑚𝑚−𝑘_𝑚+1 are previously known admissible exponents; then the exponent ^𝑚_𝑖=1𝑘_𝑖𝑚−𝑘_𝑖+1=^𝑚_𝑖=1𝑘_𝑖−1𝑚−^𝑚_𝑖=1𝑘_𝑖 (12) of 𝑓(𝑓(𝑎^{𝑘1𝑚−𝑘1+1}),𝑓(𝑎^{𝑘2𝑚−𝑘2+1}),…,𝑓(𝑎^{𝑘𝑚𝑚−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 𝑓𝑎,𝑥₁,…,𝑥_𝑚−1=𝑓𝑥₁,𝑎,…,𝑥_𝑚−1=𝑓𝑥₁,…,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 one which satisfies 𝑓(𝑎^{𝑘𝑚−𝑘+1})={0} for some integer 𝑘 greater than 0.

Theorem 3. An algebraic hypersystem 𝐻 which is commutative, associative, regular, and has a 0 with respect to an 𝑚-ary hyperoperation𝑓possesses no nilpotent element other than 0.

Proof. For all 0≠𝑎∈𝐻, let [𝑎] denote the subhypersystem of 𝐻 generated by 𝑎, which may be inductively defined as follows: (1)𝑎∈[𝑎](2)𝑓(𝑎^𝑚)⊆[𝑎](3)whenever 𝑓(𝑎^𝑛1),…,𝑓(𝑎^𝑛𝑚)⊆[𝑎], then also 𝑓(𝑎^{𝑛1+⋯+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(𝑎^𝑚)≠{0}. For, if 𝑓(𝑎^𝑚)={0}, then by virtue of the associativity, commutativity, and regularity of the given hyperoperation, there exist 𝑥₁,…,𝑥_𝑚−1∈𝐻 such that 𝑎∈𝑓(𝑎,𝑥₁,𝑎,…,𝑓(…,𝑥_𝑚−1,𝑎))=𝑓(𝑓(𝑎,𝑎,…,𝑎),𝑥₁,…,𝑥_𝑚−1)=𝑓(𝑓(𝑎^𝑚),𝑥₁,…,𝑥_𝑚−1)=𝑓(0,𝑥₁,…,𝑥_𝑚−1)={0} contrary to (1).(3)We now show that if 𝑓(𝑎^𝑛1),…,𝑓(𝑎^𝑛𝑚) are all nonzero elements of [𝑎], then 𝑓(𝑓(𝑎^𝑛1),𝑓(𝑎^𝑛2),…,𝑓(𝑎^𝑛𝑚))=𝑓(𝑎^{𝑛1+𝑛2+⋯+𝑛𝑚})≠{0}. Suppose 𝑓(𝑎^{𝑛1+𝑛2+⋯+𝑛𝑚})={0}. Then by the above remark, we have 𝑛_𝑖=𝑘_𝑖𝑚−𝑘_𝑖+1for𝑖=1,2,…,𝑚. (14)Since 𝑓 is commutative, it may be assumed without loss of generality that 𝑛₁=max_𝑖𝑛_𝑖. Then 𝑚𝑛_𝑖=^𝑚_𝑖=1𝑛_𝑖+𝑚𝑛₁−^𝑚_𝑖=1𝑛_𝑖=^𝑚_𝑖=1𝑛_𝑖+^𝑚_𝑖=1𝑛_𝑖−𝑛_𝑖=^𝑚_𝑖=1𝑛_𝑖+^𝑚_𝑖=1𝑘₁−𝑘_𝑖𝑚−𝑘₁−𝑘_𝑖=^𝑚_𝑖=1𝑛_𝑖+^𝑚_𝑖=1𝑘₁−𝑘_𝑖𝑚−^𝑚_𝑖=1𝑘₁−𝑘_𝑖+()=−𝑚+2𝑚−2_𝑚𝑖=1𝑛_𝑖+^𝑚_𝑖=1𝑘₁−𝑘_𝑖𝑚−−1_𝑚𝑖=1𝑘₁−𝑘_𝑖+()=−2𝑚+2_𝑚𝑖=1𝑛_𝑖+𝑝+(𝑚−2), (15) where 𝑝 is an admissible exponent. Hence, by associativity, commutativity, and regularity of the hyperoperation 𝑓, there exist 𝑥₁,𝑥₂,…,𝑥_𝑚−1∈𝐻 such that {0}≠𝑓𝑎^𝑛1=𝑓𝑓𝑎^𝑛1,𝑥₁,𝑓𝑎^𝑛1,…,𝑓…,𝑥_𝑚−1,𝑓𝑎^𝑛1=𝑓𝑓𝑎^𝑚𝑛1,𝑥₁,…,𝑥_𝑚−1=𝑓𝑓𝑎^{𝑛1+𝑛2+⋯+𝑛𝑚+𝑝+(𝑚−2)},𝑥₁,…,𝑥_𝑚−1=𝑓𝑓𝑎^{𝑛1+𝑛2+⋯+𝑛𝑚},𝑓𝑓𝑎^𝑝,𝑎,𝑎,…,𝑎,𝑥₁,𝑥₂,…,𝑥_𝑚−1=𝑓0,𝑓𝑓𝑎^𝑝,𝑎,𝑎,…,𝑎,𝑥₁,𝑥₂,…,𝑥_𝑚−1={0}, (16) a contradiction. Thus, every element of [𝑎] is nonzero and the conclusion follows.

Acknowledgments

Kostaq Hila wishes to express his warmest thanks to Professor Apostolos Thoma and the Department of Mathematics, University of Ioannina (Greece) for their hospitality during the author’s postdoctoral scholarship supported by the State Scholarships Foundation of Hellenic Republic.