Semilattices of rectangular bands and groups of order two

Filomat, 2010

Semigroups having a decomposition into a band of semigroups have been studied in many papers. In the present paper we give characterizations of various special types of bands of ?-semigroups and semilattices of matrices of ?- semigroups.

Semigroup Forum, 2010

In this paper we define the radical ϱ k (k∈Z +) of a relation ϱ on an arbitrary semigroup. Also, we define various types of k-regularity of semigroups and various types of k-Archimedness of semigroups. Using these notions we describe the structure of semigroups in which ρ k is a band (semilattice) congruence for some Green’s relation.

Transactions of the American Mathematical Society, 1970

Presented in this paper is a method of constructing a compact semigroup 5 from a compact semilattice Xand a compact semigroup T having idempotents contained in X. The notions of semigroups (straight) through chains and (straight) through semilattices are introduced. It is shown that the notion of a semigroup through a chain is equivalent to that of a generalized hormos. Universal objects are obtained in several categories including the category of clans straight through a chain and the category of clans straight through a semilattice relative to a chain. An example is given of a nonabelian clan S with abelian set of idempotents E such that S is minimal (as a clan) about E.

Glasgow Mathematical Journal, 1971

Let Q(S) denote the maximal right quotient semigroup of the semigroup S as defined in [4]. In this paper, we initiate a study of Q(S) when S is a semilattice of groups. A structure theorem for such semigroups is given by Theorem 4.11 of [2].

AKCE International Journal of Graphs and Combinatorics, 2019

The Square element graph over a semigroup S is a simple undirected graph Sq(S) whose vertex set consists precisely of all the non-zero elements of S, and two vertices a, b are adjacent if and only if either ab or ba belongs to the set {t 2 : t ∈ S} \ {1}, where 1 is the identity of the semigroup (if it exists). In this paper, we study the various properties of Sq(S). In particular, we concentrate on square element graphs over three important classes of semigroups. First, we consider the semigroup Ω n formed by the ideals of Z n. Afterwards, we consider the symmetric groups S n and the dihedral groups D n. For each type of semigroups mentioned, we look into the structural and other graph-theoretic properties of the corresponding square element graphs. c

Filomat, 2012

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.

Semigroup Forum, 1996

General decomposition problems hold a central place in the general structure theory of semigroups, as they look for different ways to break a semigroup into parts, with as simple a structure as possible, in order to examine these parts in detail, as well as the relationships between the parts within the whole semigroup. The main problem is to determine whether the greatest decomposition of a given type exists, the decomposition having the finest components, and to give a characterization and construction of this greatest decomposition. Another important issue is whether a given type of decomposition is atomic, in the sense that the components of the greatest decomposition of the given type cannot further be broken down by decomposition of the same type. In semigroup theory only five types of atomic decompositions are known so far. The atomicity of semilattice decompositions was proved by Tamura [Osaka Math. J. 8 (1956) 243-261], of ordinal decompositions by Lyapin [Semigroups, Fizmatgiz, Moscow, 1960], of the so-called U-decompositions by Shevrin [Dokl. Akad. Nauk SSSR 138 (1961) 796-798], of orthogonal decompositions by Bogdanović andĆirić [Israel J. Math 90 (1995) 423-428], whereas the atomicity of subdirect decompositions follows from a more general result of universal algebra proved by Birkhoff [Bull. AMS 50 (1944) 764-768]. Semilattice decompositions of semigroups were first defined and studied by A. H. Clifford [Annals of Math. 42 (1941) 1037-1049]. Later T. Tamura and N. Kimura [Kodai Math. Sem. Rep. 4 (1954) 109-112] proved the existence of the greatest semilattice decomposition of an arbitrary semigroup, and as we have already noted, while T. Tamura [Osaka Math. J. 8 (1956) 243-261] proved the atomicity of semilattice decompositions. The theory of the greatest semilattice decompositions of semigroups has been developed from the middle of the 1950s to the middle i ii PREFACE of the 1970s by T. Tamura, M. S. Putcha, M. Petrich, and others. For a long time after that there were no new results in this area. In the mid of 1990s, the authors of this book initiated the further development of this theory by introducing completely new ideas and methodology. The purpose of this book is to give an overview of the main results on semilattice decompositions of semigroups which appeared in the last 15 years, as well as to connect them with earlier results.

2015

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.

2016

This paper is for one part a generalization of some results obtained by Miyuki Yamada [20] in the case of binary semigroups to ternary semigroups. We prove analogous of almost all the results previously cited. We prove in particular that the set of the idempotents in regular ternary semigroup is a band (that is, a semigroup). In a second part we continue our investigations started in [13; 14] on these semigroups, as on the structure of the set E(S) of idempotents of the ternary semigroup S. The particular case of ternary inverse semigroup has been studied and a relationship between the existence of idempotents and the inverse elements has been caracterized. The documents [5]; [9] and [10] have been intensively used. We asked two questions and the answer for the second one will be the subject of a forcoming paper. We use many references in our work the most important are those used as bibliography.

2009

Semilattice-ordered semigroup is important algebraic structure. It is equipped with the natural defined positive quasi-antiorder relation.