Fuzzy ∗–ideals and their applications in characterizing abundance and regularity of a semigroup

In this paper, the notion of a fuzzy *–ideal of a semigroup is introduced by exploiting generalized Green’s relations L * and R * , and some characterizations of fuzzy *–ideals on an arbitrary semigroup are obtained. Our main purpose is to establish the relationship between fuzzy *–ideals and abundance for an arbitrary semigroup. As an application of our results, we also give some new necessary and sufficient conditions for an arbitrary semigroup to be regular and inverse, respectively.

A new definition of conjugacy for semigroups

The conjugacy relation plays an important role in group theory. If [Formula: see text] and [Formula: see text] are elements of a group [Formula: see text], [Formula: see text] is conjugate to [Formula: see text] if [Formula: see text] for some [Formula: see text]. The group conjugacy extends to inverse semigroups in a natural way: for [Formula: see text] and [Formula: see text] in an inverse semigroup [Formula: see text], [Formula: see text] is conjugate to [Formula: see text] if [Formula: see text] and [Formula: see text] for some [Formula: see text]. In this paper, we define a conjugacy for an arbitrary semigroup [Formula: see text] that reduces to the inverse semigroup conjugacy if [Formula: see text] is an inverse semigroup. (None of the existing notions of conjugacy for semigroups has this property.) We compare our new notion of conjugacy with existing definitions, characterize the conjugacy in basic transformation semigroups and their ideals using the representation of transformations as directed graphs, and determine the number of conjugacy classes in these semigroups.

Bands of η-simple semigroups

In this paper on an arbitrary semigroup we define a few different types of relations and its congruence extensions. Also, we describe the structure of semigroups in which these relations are band congruences. The components of such obtained band decompositions are in some sense simple semigroups.

Almost Commutative Semigroups

The commutativity degree of groups and rings has been studied by certain authors since 1973, and the main result obtained is [Formula: see text], where Pr (A) is the commutativity degree of a non-abelian group (or ring) A. Verifying this inequality for an arbitrary semigroup A is a natural question, and in this paper, by presenting an infinite class of finite non-commutative semigroups, we prove that the commutativity degree may be arbitrarily close to 1. We name this class of semigroups the almost commutative or approximately abelian semigroups.

EQUATIONAL BASES FOR SOME 0-DIRECT UNIONS OF SEMIG OUPS

In this paper,we investigate identities satis .ed by 0-direct unions of a semigroup with its anti-isomorphic copy,which serve as the standard tool for showing that an arbitrary semigroup can be embedded in (a semigroup reduct of)an involution semigroup.We show that,given the set of semigroup identities they satisfy,the involution de .ned on such 0-direct unions can be captured by only two additional identities involving the unary operation symbol.As a corollary of a result on .niteness of equational bases for such involution semigroups,we present an involution semigroup (which is,however,not an inverse one)consisting of 13 elements and not having a .nite equational basis.

On joins with group congruences

Let be an arbitrary semigroup. A congruence γ on is a group congruence if /γ is a group. The set of group congruences on is non-empty since × is a group congruence. The lattice of congruences on a semigroup will be denoted by () and the set of group congruences on will be denoted by (). If () is a lattice then it is modular and γ ∨ ρ = γ ο ρ = ρ ο γ for all γ, ρ ε (). The main result is that γ ν ρ = γ ο ρ ο γ for any γ ε () and ρ ε () (whence every element of the set () is dually right modular in (). This result has appeared, for particular classes of semigroups, many times in the literature. Also γ ν ρ = γ ο ρ ο γ = ρ ο γ ο ρ for all γ, ρ ε () which is similar to the well known result for the join of congruences on a group. Furthermore, if γ ∩ ρ ε () then γ ν ρ = γ ο ρ = ρ ο γ.

REPRESENTATIONS OF SEMIGROUPS BY TRANSFORMATIONS AND THE CONGRUENCE LATTICE OF AN EVENTUALLY REGULAR SEMIGROUP

On any eventually regular semigroup S, congruences ν, μL, μR, μ, K, KL, KR, ζ are introduced which are the greatest congruences over: nil-extensions (n.e.) of completely simple semigroups, n.e. of left groups, n.e. of right groups, n.e. of groups, n.e. of rectangular bands, n.e. of left zero semigroups, n.e. of right zero semigroups, nil-semigroups, respectively. Each of these congruences is induced by a certain representation of S which is defined on an arbitrary semigroup. These congruences play an important role in the study of lattices of varieties, pseudovarieties and existence varieties. The investigation also leads to eight complete congruences U, Tt, Tr, T, K, Kl, Kr, Z on the congruence lattice Con (S) of S.

On variants of a semigroup

If S is a (multiplicative) Semigroup and a ∈ S, the binary operation ∘ defined on the set S by x ∘ y = x a y is associative and the resulting semigroup is called a variant of S. We study the congruence a defined on S by saying that two elements are α-related if and only if they determine the same variant of S. Certain quotients of variants are used to provide an arbitrary semigroup with a generalised local structure. The variant formulation of Nambooripad's partial order on a regular semigroup is used to show that the order possesses a certain property (involving D-equivalence).

On the radical of semigroup algebras satisfying polynomial identities

In this paper we will be concerned with the problem of describing the Jacobson radical of the semigroup algebraK[S] of an arbitrary semigroupSover a fieldKin the case where this algebra satisfies a polynomial identity. Recently, Munn characterized the radical of commutative semigroup algebras [9]. The key to his result was to show that, in this situation, the radical must be a nilideal. We are going to extend the latter to the case of PI-semigroup algebras. Further, we characterize the radical by means of the properties ofSor, more precisely, by some groups derived fromS. For this purpose we will exploit earlier results leading towards a characterization of semigroup algebras satisfying polynomial identities [5], [15], which generalized the well known case of group algebras (cf. [13], chap. 5).