Regular Semigroups Research Articles

Regular semigroups weakly generated by set

AbstractWe prove the existence and uniqueness (up to isomorphism) of a regular semigroup $${{\,\textrm{F}\,}}(X)$$ F ( X ) weakly generated by a set X of elements such that all other regular semigroups weakly generated by X are homomorphic images of $${{\,\textrm{F}\,}}(X)$$ F ( X ) . The semigroup $${{\,\textrm{F}\,}}(X)$$ F ( X ) is introduced by a presentation and the word problem for that presentation is solved. The structure of the semigroup $${{\,\textrm{F}\,}}(X)$$ F ( X ) is also studied.

On Some Classes of Regular Semigroups

In this paper we study and analyze the relationships between some special classes of regular semigroups, namely: Generalized inverse semigroups, Orthodox semigroups and locally inverse semigroups based on their idempotent properties. We then use an example to prove that every locally inverse semigroup is not a generalized inverse semigroup, thus validating the statement; “A generalized inverse semigroup has the property of being locally inverse. However, not every locally inverse semigroup is a generalized inverse semigroup.”

Normed-Bifuzzy Valued-Ideals of Semigroups

As concerning the views of T norms and T conorms, the intent of article is to define and probe the fuzzy semigroups, fuzzy ideals, fuzzy bi-ideals, bifuzzy subsemigroups, bifuzzy ideals, bifuzzy bi-ideals, fuzzy (1, 2)-ideals and bifuzzy (1, 2)-ideals in any given semigroup. Also, we indicate and study their basic properties of them in completely regular semigroups. Finally, we extend these concepts and so characterize (pre)image of them in semigroup homomorphisms.

Heights of posets associated with Green's relations on semigroups

Given a semigroup S, for each Green's relation K∈{L,R,J,H} on S, the K-height of S, denoted by HK(S), is the height of the poset of K-classes of S. More precisely, if there is a finite bound on the sizes of chains of K-classes of S, then HK(S) is defined as the maximum size of such a chain; otherwise, we say that S has infinite K-height. We discuss the relationships between these four K-heights. The main results concern the class of stable semigroups, which includes all finite semigroups. In particular, we prove that a stable semigroup has finite L-height if and only if it has finite R-height if and only if it has finite J-height. In fact, for a stable semigroup S, if HL(S)=n then HR(S)≤2n−1 and HJ(S)≤2n−1, and we exhibit a family of examples to prove that these bounds are sharp. Furthermore, we prove that if 2≤HL(S)<∞ and 2≤HR(S)<∞, then HJ(S)≤HL(S)+HR(S)−2. We also show that for each n∈N there exists a semigroup S such that HL(S)=HR(S)=2n+n−3 and HJ(S)=2n+1−4. By way of contrast, we prove that for a regular semigroup the L-, R- and H-heights coincide with each other, and are greater or equal to the J-height. Moreover, in a stable, regular semigroup the L-, R-, H- and J-heights are all equal.

The number of spanning trees of E-extended power graphs associated with finite semigroups

We introduce a new kind of graph built from finite semigroups that we refer to as E-extended power graphs. An E-extended power graph ΓS of a finite semigroup S can be considered as the union of the power graph P(S) of S and the tree generated by E(S), where E(S) is the set of idempotents of S. Actually, ΓS is a minimal connected graph with vertex set S and containing P(S). In particular, if S is a finite semigroup with a unique idempotent we have ΓS=P(S). We give the number of spanning trees of an E-extended power graph of a finite semigroup, in particular, (Zn,⋅). Finally, as an application we discuss the number of spanning trees of an E-extended power graph of completely simple semigroups and completely regular semigroups.

Finite regular semigroups with permutations that map elements to inverses

We give an account on what is known on the subject of permutation matchings, which are bijections of a finite regular semigroup that map each element to one of its inverses. This includes partial solutions to some open questions, including a related novel combinatorial problem.

A groupoid approach to regular ⁎-semigroups

In this paper we develop a new groupoid-based structure theory for the class of regular ⁎-semigroups. This class occupies something of a ‘sweet spot’ between the important classes of inverse and regular semigroups, and contains many natural examples. Some of the most significant families include the partition, Brauer and Temperley-Lieb monoids, among other diagram monoids.Our main result is that the category of regular ⁎-semigroups is isomorphic to the category of so-called ‘chained projection groupoids’. Such a groupoid is in fact a triple (P,G,ε), where:•P is a projection algebra (in the sense of Imaoka and Jones),•G is an ordered groupoid with object set P, and•ε:C→G is a special functor, where C is a certain natural ‘chain groupoid’ constructed from P.Roughly speaking: the groupoid G=G(S) remembers only the ‘easy’ products in a regular ⁎-semigroup S; the projection algebra P=P(S) remembers only the ‘conjugation action’ of the projections of S; and the functor ε=ε(S) tells us how G and P ‘fit together’ in order to recover the entire structure of S. In this way, we obtain the first completely general structure theorem for regular ⁎-semigroups.As a consequence of our main result, we give a new proof of the celebrated Ehresmann–Schein–Nambooripad Theorem, which establishes an isomorphism between the categories of inverse semigroups and inductive groupoids. Other applications will be given in future works.We consider several examples along the way, and pose a number of problems that we believe are worthy of further attention.

Some Properties of Fuzzy Ideals of Unit Regular Semigroups

This paper introduces the concept of unit-regular fuzzy ideals of semi groups, which is a generalization of regular fuzzy ideals. Several of their significant related properties have been examined. According to the unit-regular semi group, the fuzzy-interior-ideal, fuzzy-ideal, fuzzy-bi-ideal, and fuzzy-generalized-bi-ideal have all been described.

Regular semigroups weakly generated by one element

In this paper we study the regular semigroups weakly generated by a single element x, that is, with no proper regular subsemigroup containing x. We show there exists a regular semigroup F_{1} weakly generated by x such that all other regular semigroups weakly generated by x are homomorphic images of F_{1}. We define F_{1} using a presentation where both sets of generators and relations are infinite. Nevertheless, the word problem for this presentation is decidable. We describe a canonical form for the congruence classes given by this presentation, and explain how to obtain it. We end the paper studying the structure of F_{1}. In particular, we show that the ‘free regular semigroup {textrm{FI}}_2 weakly generated by two idempotents’ is isomorphic to a regular subsemigroup of F_{1} weakly generated by {xx',x'x}.

Hybrid interior ideals and hybrid bi-ideals in ternary semigroups

This article explores the concept of hybrid interior ideals and the hybrid characteristic interior ideals of a ternary semigroup. We establish some equivalent requirements for a hybrid structure to be a hybrid interior ideal of a ternary semigroup. Also, we show that for a regular semigroup and an intra-regular semigroup, hybrid interior ideals and hybrid ideals coincide. Furthermore, the notion of a hybrid bi-ideal of a ternary semigroup is introduced, and some of its key characteristics are explored.

Cross connection compatible discrete normal categories

The concept of normal category was introduced by Nambooripad in his theory of cross connections for describing the structure of regular semigroups. These are essentially the categories of principal left ideals and the categories of principal right ideals of regular semigroups with appropriate morphisms. These categories also have an associated partial order on the set of objects. Discrete normal categories are normal categories in which the partial order on the object set is the identity relation. A cross connection relates two normal categories [Formula: see text] and [Formula: see text] and produces a regular semigroup [Formula: see text] such that one is the category of principal left ideals and the other the category of principal right ideals of the semigroup [Formula: see text]. This paper considers the question of compatibility of two normal categories in order that they can be connected by a cross connection. It is known that given any normal category [Formula: see text] there is a normal category [Formula: see text] called the normal dual of [Formula: see text] such that [Formula: see text] is compatible with [Formula: see text]. The question of finding all normal categories [Formula: see text] that are compatible with a given normal category [Formula: see text] remains unsettled. Such categories [Formula: see text] are called compatible duals of [Formula: see text]. For the class of discrete normal categories the characterization of all compatible duals has been done here.

Regular semigroups weakly generated by idempotents

A regular semigroup is weakly generated by a set [Formula: see text] if it has no proper regular subsemigroup containing [Formula: see text]. In this paper, we study the regular semigroups weakly generated by idempotents. We show there exists a regular semigroup [Formula: see text] weakly generated by [Formula: see text] idempotents such that all other regular semigroups weakly generated by [Formula: see text] idempotents are homomorphic images of [Formula: see text]. The semigroup [Formula: see text] is defined by a presentation [Formula: see text] and its structure is studied. Although each of the sets [Formula: see text], [Formula: see text], and [Formula: see text] is infinite for [Formula: see text], we show that the word problem is decidable as each congruence class has a “canonical form”. If [Formula: see text] denotes [Formula: see text] for [Formula: see text], we prove also that [Formula: see text] contains copies of all [Formula: see text] as subsemigroups. As a consequence, we conclude that (i) all regular semigroups weakly generated by a finite set of idempotents, which include all finitely idempotent generated regular semigroups, strongly divide [Formula: see text]; and (ii) all finite semigroups divide [Formula: see text].

Normal category of principal left ideals of bands

The principal left ideals of a regular semigroup have been abstractly described by Nambooripad as normal categories. A semigroup [Formula: see text] is said to be a band if [Formula: see text] for all [Formula: see text]. Since bands are regular semigroups, the category of principal left ideals is a normal category. This paper characterizes the normal categories which arise as categories of principal left ideals of bands.

Saturated (n, m)-Regular Semigroups

The aim of this paper is to determine several saturated classes of structurally regular semigroups. First, we show that structurally (n,m)-regular semigroups are saturated in a subclass of semigroups for any pair (n,m) of positive integers. We also demonstrate that, for all positive integers n and k with 1≤k≤n, the variety of structurally (0,n)-left seminormal bands is saturated in the variety of structurally (0,k)-bands. As a result, in the category of structurally (0,k)-bands, epis from structurally (0,n)-left seminormal bands is onto.

On a Generalization of Fuzzy Filters of Ordered Semigroups

The aim of this paper is to generalize the results related to fuzzy filters of ordered semigroups. In this regard, the concept of fuzzy [Formula: see text]-filters of ordered semigroups is initiated, nontrivial examples are presented, and its properties are investigated. It is known that a fuzzy set of an ordered semigroup is a fuzzy filter if and only its complement is a fuzzy prime ideal. As a generalization of the latter result, it is proved that a fuzzy set of an ordered semigroup is a fuzzy [Formula: see text]-filter if and only if its complement is a prime generalized fuzzy [Formula: see text]-ideal. Furthermore, it is shown that in an [Formula: see text]-regular semigroup, fuzzy [Formula: see text]-filters and fuzzy bi-filters coincide.

MULTIPLICATION TABLES AND WORD-HYPERBOLICITY IN FREE PRODUCTS OF SEMIGROUPS, MONOIDS AND GROUPS

AbstractThis article studies the properties of word-hyperbolic semigroups and monoids, that is, those having context-free multiplication tables with respect to a regular combing, as defined by Duncan and Gilman [‘Word hyperbolic semigroups’,Math. Proc. Cambridge Philos. Soc.136(3) (2004), 513–524]. In particular, the preservation of word-hyperbolicity under taking free products is considered. Under mild conditions on the semigroups involved, satisfied, for example, by monoids or regular semigroups, we prove that the semigroup free product of two word-hyperbolic semigroups is again word-hyperbolic. Analogously, with a mild condition on the uniqueness of representation for the identity element, satisfied, for example, by groups, we prove that the monoid free product of two word-hyperbolic monoids is word-hyperbolic. The methods are language-theoretically general, and apply equally well to semigroups, monoids or groups with a$\mathbf {C}$-multiplication table, where$\mathbf {C}$is any reversal-closed super-$\operatorname {\mathrm {AFL}}$. In particular, we deduce that the free product of two groups with$\mathbf {ET0L}$with respect to indexed multiplication tables again has an$\mathbf {ET0L}$with respect to an indexed multiplication table.

On Interval Valued Fuzzy Bi-interior Ideals in Semigroups

In this paper, we study the concept of an interval valued fuzzy bi-interior ideal. We investigate the properties of an interval valued fuzzy bi-interior ideal in semigroups. We characterize a regular semigroup in terms of an interval valued fuzzy bi-interior ideal.

Cross-connection structure of locally inverse semigroups

Locally inverse semigroups are regular semigroups whose idempotents form pseudo-semilattices. We characterize the categories that correspond to locally inverse semigroups in the realm of Nambooripad’s cross-connection theory. Further, we specialize our cross-connection description of locally inverse semigroups to inverse semigroups and completely [Formula: see text]-simple semigroups, obtaining structure theorems for these classes. In particular, we show that the structure theorem for inverse semigroups can be obtained using only one category, quite analogous to the Ehresmann–Schein–Nambooripad Theorem; for completely [Formula: see text]-simple semigroups, we show that cross-connections coincide with structure matrices, thus recovering the Rees Theorem by categorical tools.

Regularity of Semigroups in Terms of Hybrid Ideals and Hybrid Bi-Ideals

In this paper, we establish some equivalent conditions for a semigroup to be regular and intra-regular, in terms of hybrid ideals and hybrid bi-ideals. We also characterize the left and right simple and the completely regular semigroups utilizing hybrid ideals and hybrid bi-ideals. We show that a semigroup S is left simple if and only if it is hybrid left simple. We also prove that if a semigroup S is intra-regular, then for each hybrid ideal ˜jµ of S, we have ˜jµ(r1r2) = ˜jµ(r2r1) for all r1, r2 ∈ S.

On spined product decompositions of completely regular semigroups

Here, we have considered two equivalence relations [Formula: see text] and [Formula: see text] that are congruences on regular orthogroups but not on completely regular semigroups, in general. The congruence openings [Formula: see text] and [Formula: see text] of [Formula: see text] and [Formula: see text], respectively, induce new subvarieties [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] for every [Formula: see text]; and are such that [Formula: see text] and [Formula: see text]. The decompositions of an arbitrary completely regular semigroup which are presented here generalize several spined product decompositions for particular classes of completely regular semigroups already given by several authors [M. Petrich and N. R. Reilly, Completely Regular Semigroup (Wiley, New York, 1999)], and thus contributes to a unification of the theory.