Structure Of Regular Semigroups Research Articles

A Study on Regular Semigroups and its Idempotents

An idempotent of a semigroup T is an element e in T such that In many semigroups, idempotents can be recognized easily. Thus it plays an important role in the structure of semigroups especially on regular semigroups. This article reviews about some research work done about the structure of regular semigroups with a special emphasis on its idempotents.

Homomorphisms and congruences on regular semigroups with associate inverse subsemigroups

An associate inverse subsemigroup of a regular semigroup S is a subsemigroup T of S containing a least associate x* of each x ∈ S, in relation to the natural partial order ≤. In [1] the authors describe the structure of regular semigroups with an associate inverse subsemigroup, satisfying two natural conditions. In this paper we describe all *-homomorphisms and all *-congruences on such semigroups.

Structure of regular semigroups with a quasi-ideal inverse transversal

Let S be a regular semigroup, and E(S) denotes the set of idempotents of S. An inverse subsemigroup S ~ of S is called an inverse transversal of S, if IS ~ n V(x) l = I for each x ~ S, where V(x) denotes the set of inverses of x. An inverse of x c S which belongs to S ~ is denoted by x ~ and x ~176 denotes (x~ ~ = (x~ -z. A subset Q of S is called a quasi-ideal of S if QSQ ~ Q. Let S be a regular semigroup with an inverse transversal S ~ that is a quasi-ideal of S. In [4], it has been shown that the set I = {e c S : ee ~ = e} and A = {f c S : fof = f} are ~-unipotent and ~-unipotent subbands respectively of S and that, if gO ~ E(SO), then g~ = g~176 for each e e I and fgO = fOgO for each f s A. By using the results, we have:

RP-dominated regular semigroups

SynopsisWe describe the structure of regular semigroups in which each element is dominated, in the Nambooripad order, by a unique maximal element that is ℋ-equivalent to a mididentity. These are precisely direct products of a rectangular band and a uniquely unit regular semigroup.

Structure of regular semigroups. I

This paper concerned with basic concepts and some results on (idempotent) semigroup satisfying the identities of three variables. The motivation of taking three for the number of variables has come from the fact that many important identities on idempotent semigroups are written by three or fewer independent variables. We consider the semigroup satisfying the property abc = ac and prove that it is left semi- normal and right quasi-normal. Again an idempotent semigroup with an identity aba = ab and aba = ba (ab = a, ab = b) is always a semilattices and normal. An idempotent semigroup is normal if and only if it is both left quasi-normal and right quasi-normal. If a semigroup is rectangular then it is left and right semi-regular. I. PRELIMINARIES AND BASIC PROPERTIES OF REGULAR SEMIGROUPS In this section we present some basic concepts of semigroups and other definitions needed for the study of this chapter and the subsequent chapters. 1.1 Definition: A semigroup (S, .) is said to be left(right) singular if it satisfies the identity ab = a (ab = b) for all a,b in S 1.2 Definition: A semigroup (S, .) is rectangular if it satisfies the identity aba = a for all a,b in S. 1.3 Definition: A semigroup (S, .) is called left(right) regular if it satisfies the identity aba = ab (aba = ba) for all a,b in S. 1.4 Definition: A semigroup (S, .) is called regular if it satisfies the identity abca = abaca for all a,b,c in S 1.5 Definition: A semigroup (S, .) is said to be total if every element of Scan be written as the product of two elements of S. i.e, S 2

The structure of regular semigroups, II: Cross-connections

The structure of regular semigroups, II: Cross-connections

Structure of regular semigroups, I fundamental regular semigroups

Structure of regular semigroups, I fundamental regular semigroups

The structure of regular semigroups, III: The reduced case

Reduced regular semigroups were characterized in the previous announcements of this sequence: up to isomorphism, they are the full regular subsemigroups of certain semigroups U built from regular partially ordered sets. We now complete this result by an explicit procedure to pick out all full regular subsemigroups of any U. This also provides a way to obtain (from partially ordered sets) all possible egg-box pictures of regular semigroups, as well as a blueprint for the multiplication in the general case.

The structure of regular semigroups, I: A representation

The structure of regular semigroups, I: A representation

The structure of regular semigroups, IV: The general case

The structure of regular semigroups, IV: The general case

Structure of regular semigroups II the general case

Structure of regular semigroups II the general case

Regular semigroups satisfying certain conditions on idempotents and ideals

The structure of regular semigroups is studied (1) whose poset of idempotents is required to be a tree or to satisfy a weaker condition concerning the behavior of idempotents in different !D-classes, or (2) all of whose ideals are categorical or satisfy a variation thereof. For this purpose the notions of Dmajorization of idempotents, where D is a ED-class, !-majorization, E-categorical ideals, and completely semisimple semigroups without contractions are introduced and several connections among them are established. Some theorems due to G. Lallement concerning subdirect products and completely regular semigroups and certain results of the author concerning completely semisimple inverse semigroups are either improved or generalized.

Note on the structure of regular semigroups

Note on the structure of regular semigroups