Classes of semigroups modulo Green’s relation H (original) (raw)
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Classes of semigroups modulo Green’s relation mathcalH\mathcal{H}mathcalH
Semigroup Forum, 2013
Inverses semigroups and orthodox semigroups are either defined in terms of inverses, or in terms of the set of idempotents E(S). In this article, we study analogs of these semigroups defined in terms of inverses modulo Green's relation H, or in terms of the set of group invertible elements H(S), that allows a study of non-regular semigroups. We then study the interplays between these new classes of semigroups, as well as with known classes of semigroups (notably inverse, orthodox and cryptic semigroups).
©Nigerian Mathematical Society VARIANT OF FINITE SYMMETRIC INVERSE SEMIGROUP
2020
In a semigroup S fix an element a ∈ S and, for all x, y ∈ S, define a binary operation ∗a on S by x ∗a y = xay, (where the juxterposition on the left denote the original semigroup operation on S). The operation ∗a is clearly associative and so S forms a new semigroup under this operation, which is denoted by S and called variant of S by a. For a finite set Xn = {1, 2, ..., n}, let In be the symmetric inverse semigroup on Xn and fix an idempotent a ∈ In. In this paper, we study the variant I n of In by a. In particular, we characterised Green’s relations and starred Green’s relations L∗, R∗ in I n and also showed that the variant semigroup I n is abundant.
Restriction and Ehresmann Semigroups
Proceedings of the International Conference on Algebra 2010 - Advances in Algebraic Structures, 2011
Inverse semigroups form a variety of unary semigroups, that is, semigroups equipped with an additional unary operation, in this case a → a −1. The theory of inverse semigroups is perhaps the best developed within semigroup theory, and relies on two factors: an inverse semigroup S is regular, and has semilattice of idempotents. Three major approaches to the structure of inverse semigroups have emerged. Effectively, they each succeed in classifying inverse semigroups via groups (or groupoids) and semilattices (or partially ordered sets). These are (a) the Ehresmann-Schein-Nambooripad characterisation of inverse semigroups in terms of inductive groupoids, (b) Munn's use of fundamental inverse semigroups and his construction of the semigroup T E from a semilattice E, and (c) McAlister's results showing on the one hand that every inverse semigroup has a proper (E-unitary) cover, and on the other, determining the structure of proper inverse semigroups in terms of groups, semilattices and partially ordered sets. The aim of this article is to explain how the above techniques, which were developed to study inverse semigroups, may be adapted for certain classes of bi-unary semigroups. The classes we consider are those of restriction and Ehresmann semigroups. The common feature is that the semigroups in each class possess a semilattice of idempotents; however, there is no assumption of regularity.
arXiv: Group Theory, 2019
The purpose of this paper is to study the generalization of inverse semigroups (without order). An ordered semigroup S is called an inverse ordered semigroup if for every a 2 S, any two inverses of a are H-related. We prove that an ordered semigroup is complete semilattice of t-simple ordered semigroups if and only if it is completely regular and inverse. Furthermore characterizations of inverse ordered semigroups have been characterized by their ordered idempotents.
Idempotents, band and Green's relations on ternary semigroups
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.
O ct 2 01 8 CONJUGACY IN INVERSE SEMIGROUPS
In a group G, elements a and b are conjugate if there exists g ∈ G such that g −1 ag = b. This conjugacy relation, which plays an important role in group theory, can be extended in a natural way to inverse semigroups: for elements a and b in an inverse semigroup S, a is conjugate to b, which we will write as a ∼ i b, if there exists g ∈ S 1 such that g −1 ag = b and gbg −1 = a. The purpose of this paper is to study the conjugacy ∼ i in several classes of inverse semigroups: symmetric inverse semigroups, free inverse semigroups, McAllister P-semigroups, factorizable inverse monoids, Clifford semigroups, the bicyclic monoid and stable inverse semigroups.
J . Semigroup Theory Appl . 2013 , 2013 : 6 ISSN 2051-2937 GREEN ’ S RELATIONS ON TERNARY SEMIGROUPS
2013
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, coding theory, topological spaces and the like. This provides sufficient motivation to researchers to review various concepts and results. The theory of ternary algebraic system was introduced by D. H. Lehmer [11]. He investigated certain ternary algebraic systems called triplexes which turn out to be commutative ternary groups. The notion of ternary semigroups was introduced by Banach S. He showed by an example that a ternary semigroup does not necessarily reduce to an ordinary semigroups. In another hand, in mathematics, Green’s relations characterise the elements of a semigroup in terms of the principal ideals they generate. John Mackintosh Howie, a prominent semigroup theorist, described this work as so all-pervading that, on encountering a new semigroup, almost the first question o...
Some first tantalizing steps into semigroup theory
Semigroup theory is a thriving field in modern abstract algebra, though perhaps not a very well-known one. In this article, we give a brief introduction to the theory of algebraic semigroups and hopefully demonstrate that it has a flavor quite different from that of group theory. In the first few sections, we build up some basic semigroup theory, always with the group analogy at the back of our minds. Then, once we have established enough theory, we break free of this restriction and see some truly “independent” semigroup theory. In the final section, we consider the application of semigroups to the study of “partial symmetries.”
Inverse semigroups with zero: covers and their structure
Journal of the Australian Mathematical Society, 1999
We obtain analogues, in the setting of semigroups with zero, of McAlister's covering theorem and the structure theorems of McAlister, O'Carroll, and Margolis and Pin. The covers come from a class C of semigroups defined by modifying one of the many characterisations of E-unitary inverse semigroups, namely, that an inverse semigroup is E-unitary if and only if it is an inverse image of an idempotent-pure homomorphism onto a group. The class C is properly contained in the class of all £*-unitary inverse semigroups introduced by Szendrei but properly contains the class of strongly categorical £*-unitary semigroups recently considered by Gomes and Howie. 1991 Mathematics subject classification (Amer. Math. Soc): primary 20M18.