Rees matrix semigroups over inverse semigroups

1986 ◽  
Vol 102 (1-2) ◽  
pp. 61-90 ◽  
Author(s):  
F. J. Pastijn ◽  
Mario Petrich
Keyword(s):  
Inverse Semigroup ◽  
Homomorphic Image ◽  
Inverse Semigroups ◽  
Rees Matrix Semigroup ◽  
Regular Semigroups ◽  
Rees Matrix Semigroups ◽  
Special Cases ◽  
Matrix Semigroup ◽  
Matrix Semigroups ◽  
Completely Simple

SynopsisA Rees matrix semigroup over an inverse semigroup contains a greatest regular subsemigroup. The regular semigroups obtained in this manner are abstractly characterized here. The greatest completely simple homomorphic image and the idempotent generated part of such semigroups are investigated. Rectangular bands of semilattices of groups and some special cases are characterized in several ways.

scholarly journals Generators and relations of Rees matrix semigroups

1999 ◽  
Vol 42 (3) ◽  
pp. 481-495 ◽  
Author(s):  
H. Ayik ◽  
N. Ruškuc
Keyword(s):  
Rees Matrix Semigroup ◽  
Finitely Generated ◽  
Finite Generation ◽  
Generators And Relations ◽  
Rees Matrix Semigroups ◽  
Finite Presentability ◽  
Matrix Semigroup ◽  
Finitely Presented ◽  
Matrix Semigroups ◽  
The Ideal

In this paper we consider finite generation and finite presentability of Rees matrix semigroups (with or without zero) over arbitrary semigroups. The main result states that a Rees matrix semigroup M[S; I, J; P] is finitely generated (respectively, finitely presented) if and only if S is finitely generated (respectively, finitely presented), and the sets I, J and S\U are finite, where U is the ideal of S generated by the entries of P.

scholarly journals Flows on Classes of Regular Semigroups and Cauchy Categories

2019 ◽  
Vol 2019 ◽  
pp. 1-9
Author(s):  
Suha Ahmed Wazzan
Keyword(s):  
General Structure ◽  
Transformation Semigroups ◽  
Full Transformation ◽  
Regular Semigroups ◽  
Rees Matrix Semigroups ◽  
Completely Simple Semigroups ◽  
Special Case ◽  
Brandt Semigroups ◽  
Matrix Semigroups ◽  
Completely Simple

We consider the structure of the flow monoid for some classes of regular semigroups (which are special case of flows on categories) and for Cauchy categories. In detail, we characterize flows for Rees matrix semigroups, rectangular bands, and full transformation semigroups and also describe the Cauchy categories for some classes of regular semigroups such as completely simple semigroups, Brandt semigroups, and rectangular bands. In fact, we obtain a general structure for the flow monoids on Cauchy categories.

scholarly journals The natural partial order on a regular semigroup

1980 ◽  
Vol 23 (3) ◽  
pp. 249-260 ◽  
Author(s):  
K. S. Subramonian Nambooripad
Keyword(s):  
Inverse Semigroup ◽  
Regular Semigroup ◽  
Partial Order ◽  
Inverse Semigroups ◽  
Natural Partial Order ◽  
Simple Description ◽  
Regular Semigroups ◽  
Simple Congruence ◽  
Pseudo Inverse ◽  
Completely Simple

It is well-known that on an inverse semigroup S the relation ≦ defined by a ≦ b if and only if aa−1 = ab−1 is a partial order (called the natural partial order) on S and that this relation is closely related to the global structure of S (cf. (1, §7.1), (10)). Our purpose here is to study a partial order on regular semigroups that coincides with the relation defined above on inverse semigroups. It is found that this relation has properties very similar to the properties of the natural partial order on inverse semigroups. However, this relation is not, in general, compatible with the multiplication in the semigroup. We show that this is true if and only if the semigroup is pseudo-inverse (cf. (8)). We also show how this relation may be used to obtain a simple description of the finest primitive congruence and the finest completely simple congruence on a regular semigroup.

scholarly journals Inverse subsemigroups of Rees matrix semigroups

1973 ◽  
Vol 9 (3) ◽  
pp. 445-463
Author(s):  
David E. Zitarelli
Keyword(s):  
Simple Semigroup ◽  
Group Structure ◽  
Structure Group ◽  
Rees Matrix Semigroup ◽  
Rees Matrix Semigroups ◽  
Inverse Subsemigroups ◽  
Matrix Semigroup ◽  
Matrix Semigroups

According to the Sees Theorem, every completely 0-simple semigroup can be represented by a Rees matrix semigroup over a group with zero. A characterization of all subsemigroups of the latter is given in terms of the structure group, structure sets, and two mappings. Next all congruences on such subsemigroups are described, along with conditions for comparability. Finally, an algorithm for computing the number of nonisomorphic inverse subsemigroups is constructed.

Rees matrix covers for a class of abundant semigroups

1987 ◽  
Vol 107 (1-2) ◽  
pp. 109-120
Author(s):  
Mark V. Lawson
Keyword(s):  
Inverse Semigroup ◽  
Structure Theory ◽  
Regular Semigroups ◽  
Rees Matrix Semigroups ◽  
Abundant Semigroups ◽  
Multiplicative Semigroups ◽  
Matrix Semigroups ◽  
Cancellative Monoids

SynopsisRecently considerable attention has been paid to the study of locally inverse regular semigroups. McAlister [14] obtained a description of such semigroups as locally isomorphic images of regular Rees matrix semigroups over an inverse semigroup. The class of abundant semigroups originally arose from ‘homological’ considerations in the theory of S-systems: they are the semigroup theoretic analogue of PP-rings. Cancellative monoids, full subsemigroups of regular semigroups as well as the multiplicative semigroups of PP rings are abundant. The aim of this paper is to show how the structure theory described above for regular semigroups may be generalised to a class of abundant semigroups.

scholarly journals Rees matrix semigroups and the regular semidirect product

2005 ◽  
Vol 79 (1) ◽  
pp. 39-60
Author(s):  
K. Auinger ◽  
M. B. Szendrei
Keyword(s):  
Semidirect Product ◽  
Inverse Semigroups ◽  
Group Variety ◽  
Rees Matrix Semigroups ◽  
Completely Simple Semigroups ◽  
Matrix Semigroups ◽  
Completely Simple

AbstractA generalization of the Pastijn product is introduced so that, on the level of e-varieties and pseudoe-varieties, this product and the regular semidirect product by completely simple semigroups ‘almost always’ coincide. This is applied to give a model of the bifree objects in every e-variety formed as a regular semidirect product of a variety of inverse semigroups by a variety of completely simple semigroups that is not a group variety.

scholarly journals Subdirectly irreducible Rees matrix semigroups

1977 ◽  
Vol 16 (3) ◽  
pp. 351-359
Author(s):  
David E. Zitarelli
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Structure Group ◽  
Subdirectly Irreducible ◽  
Rees Matrix Semigroup ◽  
Rees Matrix Semigroups ◽  
Necessary And Sufficient ◽  
Matrix Semigroup ◽  
Matrix Semigroups

Minimal congruences on a Rees matrix semigroup S having at least one proper congruence are described. Necessary and sufficient conditions for S to te subdirectly irreducible are given in two cases according to whether the structure group of S is trivial.

scholarly journals Inverse semigroups and their natural order

1978 ◽  
Vol 19 (1) ◽  
pp. 59-65 ◽  
Author(s):  
H. Mitsch
Keyword(s):  
Inverse Semigroup ◽  
Partial Order ◽  
Homomorphic Image ◽  
Point Of View ◽  
Inverse Semigroups ◽  
Natural Order ◽  
Theoretical Point ◽  
Trivial Group

The natural order of an inverse semigroup defined by a ≤ b ⇔ a′b = a′a has turned out to be of great importance in describing the structure of it. In this paper an order-theoretical point of view is adopted to characterise inverse semigroups. A complete description is given according to the type of partial order an arbitrary inverse semigroup S can possibly admit: a least element of (S, ≤) is shown to be the zero of (S, ·); the existence of a greatest element is equivalent to the fact, that (S, ·) is a semilattice; (S, ≤) is directed downwards, if and only if S admits only the trivial group-homomorphic image; (S, ≤) is totally ordered, if and only if for all a, b ∈ S, either ab = ba = a or ab = ba = b; a finite inverse semigroup is a lattice, if and only if it admits a greatest element. Finally formulas concerning the inverse of a supremum or an infimum, if it exists, are derived, and right-distributivity and left-distributivity of multiplication with respect to union and intersection are shown to be equivalent.

scholarly journals On bisimple semigroups generated by a finite number of idempotents

1982 ◽  
Vol 33 (1) ◽  
pp. 92-101 ◽  
Author(s):  
Karl Byleen
Keyword(s):  
Finite Number ◽  
Partial Order ◽  
Natural Partial Order ◽  
Rees Matrix Semigroups ◽  
Matrix Semigroups ◽  
Completely Simple

AbstractNon-completely simple bisimple semigroups S which are generated by a finite number of idempotents are studied by means of Rees matrix semigroups over local submonoids eSe, e = e2 ∈ S. If under the natural partial order on the set Es of idempotents of such a semigroup S the sets ω(e) = {ƒ ∈ Es: ƒ ≤ e} for each e ∈ Es are well-ordered, then S is shown to contain a subsemigroup isomorphic to Sp4, the fundamental four-spiral semigroup. A non-completely simple hisimple semigroup is constructed which is generated by 5 idempotents but which does not contain a subsemigroup isomorphic to Sp4.

Inverse semigroups generated by a pair of subgroups

1977 ◽  
Vol 77 (1-2) ◽  
pp. 9-22 ◽  
Author(s):  
D. B. McAlister
Keyword(s):  
Inverse Semigroup ◽  
Free Product ◽  
Homomorphic Image ◽  
Main Part ◽  
Structure Theorem ◽  
Inverse Semigroups ◽  
Infinite Cyclic Group ◽  
The Third ◽  
Group A ◽  
Maximum Group

SynopsisThe aim of this paper is to describe the free product of a pair G, H of groups in the category of inverse semigroups. Since any inverse semigroup generated by G and H is a homomorphic image of this semigroup, this paper can be regarded as asking how large a subcategory, of the category of inverse semigroups, is the category of groups? In this light, we show that every countable inverse semigroup is a homomorphic image of an inverse subsemigroup of the free product of two copies of the infinite cyclic group. A similar result can be obtained for arbitrary cardinalities. Hence, the category of inverse semigroups is generated, using algebraic constructions by the subcategory of groups.The main part of the paper is concerned with obtaining the structure of the free product G inv H, of two groups G, H in the category of inverse semigroups. It is shown in section 1 that G inv H is E-unitary; thus G inv H can be described in terms of its maximum group homomorphic image G gp H, the free product of G and H in the category of groups, and its semilattice of idempotents. The second section considers some properties of the semilattice of idempotents while the third applies these to obtain a representation of G inv H which is faithful except when one group is a non-trivial finite group and the other is trivial. This representation is used in section 4 to give a structure theorem for G inv H. In this section, too, the result described in the first paragraph is proved. The last section, section 5, consists of examples.

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