Matrix Semigroups Research Articles

On the representations of the full matrix semigroup on homogeneous polynomials

This paper is the successor of [ 11. In the closing theorem there we determined the composition factors of w” for a regular I [ 1, 6.11. Here we are going to describe the structure of WA in the singular case. The second theme of this paper is the criterion for distributivity of the M -modules WCd’ Since the necessary conditions for that established in [ 1: 8.51 limit d to’d <p( p + 1) we just have to sort out the latter case. This will be done in the third section of this paper. For the convenience of the reader we recall some notation, terms, and definitions used in [ 11.

Infinite iteration of matrix semigroups I. Structure theorem for torsion semigroups

Given these conditions, we must precisely define EGC, special, and line. We begin with EGC. If X is a finite nonempty set X*, the free monoid generated by X, is by definition EGC because, intuitively, concatenation of two finite strings is a transparent multiplication. However, subsemigroups, and especially homomorphisms of EGCs, need not be EGCs. Furthermore, if S is EGC, and some “simple” operations applied to S gives S’, it is “reasonable” to say S’ is also EGC. Thus, by definition, if S is EGC and R: B x A + S, then S’ = R(S) (the Rees’ matrix semigroup over S) (see Definition 4.16), is also EGC. Note that the local structure, e.g., Green relations, egg box pictures, etc., of R(S) can be “easily” computed from the local structure of S and from R. For similar reasons, if 2 and S are EGC, R: B x A + S, (Z, A) and (B, Z) are actions, and R(bz, a) = R(b, za) for all b E B, z E Z, a E A, then, by definition, Z + R(S) (see Definition 4.16) is also EGC. 422 0021-8693186 $3.00

An Extension of a result of Csiszar

We extend the results of Csiszar (Z. Wahr. 5(1966) 279‐295) to a topological semigroup S. Let μ be a measure defined on S. We consider the value of . First. we show that the value of α is either zero or one. If α = 1, we show that there exists a sequence of elements {an} In S such that converges vaguely to a probability measure where δ denotes point mass. In particular, we apply the results to inverse and matrix semigroups.

Rees matrix semigroups over inverse semigroups

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.

Regular expansions of completely simple semigroups

Corresponding to each completely simple semigroup S a semigroup P( S) is constructed. The semigroup P( S) is a perfect rectangular band of inverse monoids (that is, P( S) a disjoint union of inverse monoids S i, λ ( i ϵ I, λ ϵ Λ) with S iλ S jμ = S iμ for all i, j ϵ I and λ, μ ϵ Λ). The functor S → P( S), Φ → P( Φ) is an expansion in the sense of Birget and Rhodes (“Almost finite expansions of arbitrary semigroups”, J. Pure Appl. Algebra 32 (1984) 239–287). A representation for P( S) as a Rees matrix semigroup M (T; I, Λ; P) over an inverse monoid T with entries from P in the group of units of T is provided. The expansion provides a construction of the free inverse semigroup and the free rectangular band of inverse semigroups when applied to the free group and the free completely simple semigroup respectively.

Strongly π-Regular Matrix Semigroups

Strongly π-Regular Matrix Semigroups

Non-modular congruence lattices of Rees matrix semigroups

Non-modular congruence lattices of Rees matrix semigroups

Strongly 𝜋-regular matrix semigroups

We prove that if S S is a strongly π \pi -regular multiplicative sub-semigroup of the matrix algebra M n ( K ) {M_n}(K) , K K being a field, then there exists a chain of ideals S 1 ◃ ⋯ ◃ S t = S {S_1} \triangleleft \cdots \triangleleft {S_t} = S such that t ≤ 2 n + 1 t \leq {2^{n + 1}} and any Rees factor semigroup S i / S i − 1 {S_i}/{S_{i - 1}} is either completely 0 0 -simple or nilpotent of index not exceeding ∏ j = 0 n ( j n ) \prod _{j = 0}^n(_j^n) . This sharpens the main result of [4], in particular solving Problem 3.9 from [3].

Group relationships and homomorphisms of Boolean matrix semigroups

The image matrix obtained from a blocked Boolean matrix gives a homomorphism provided that the product of any two nonzero blocks is nonzero. All maximal nonzero Boolean matrix semigroups are found. Those invariant under conjugations by a permutation matrix have the form G i = { A: the sum of any ith rows of A (or all rows) has at least i ones (or all ones)}.

Rees matrix covers for regular semigroups

By a local submonoid, of a regular semigroup S, we mean a subset of the form eSe, where e is an idempotent of S. Many classes of regular semigroups can be defined in terms of properties of their local submonoids. For example, rectangular bands can be characterized as those regular semigroups all of whose local submonoids are trivial while completely simple semigroups are those whose local submonoids are groups. We say that a regular semigroup has a property ‘3 locally, or is a locally 5?7 semigroup, if each local submonoid of S has property %Y. In a previous paper we showed that a regular semigroup is locally inverse if and only if it is an image, by a homomorphism which is one to one on local submonoids, of a regular Rees matrix semigroup over an inverse semigroup. In this paper we extend that result to various other classes of regular semigroups. In particular, we show the analog of this result for locally E-solid semigroups. (A regular semigroup is E-solid if the subsemigroup generated by its idempotents is a union of groups.) The class of locally E-solid regular semigroups is extremely extensive. It includes almost all classes of regular semigroups which have been studied from a structural point of view since inverse semigroups, orthodox semigroups, unions of groups semigroups and their localizations all belong to this class. The first section of the paper contains preliminary results which we shall require later in the paper. Section 2 describes a general procedure for constructing Rees matrix covers for regular semigroups which is applied in subsequent sections to obtain Rees matrix covers for special classes of regular semigroups. These classes include locally E-solid regular semigroups, locally orthodox semigroups and locally Sunipotent semigroups. The covering theorems on these classes of regular semigroups can be obtained directly through a painstaking analaysis of special cases. We have, however, chosen to obtain them as applications of the covering theorem for locally inverse semigroups which was proved earlier [ 81 by making use of recent interesting results of Hall concerning the relationship between locally E-solid

Completely zero-simple semigroups generated by nilpotent elements

For a completely 0-simple semigroup, Howie [2] has investigated the subsemigroup generated by the idempotents. Here we determine those elements of such a semigroup which are generated by the set of nilpotent elements and hence we derive a condition for a completely 0-simple semigroup to be nilpotent generated. This condition is purely combinatorial, in terms of the structure of the graph associated with the semigroup, and it includes the case of a non-regular Rees matrix semigroup.

Embedding any countable semigroup in a 2-generated bisimple monoid

G. B. Preston [10] proved that any semigroup can be embedded in a bisimple monoid. If S is a countable semigroup, his constructive proof yields a bisimple monoid which is also countable, but not necessarily finitely generated. The main result of this paper is that any countable semigroup can be embedded in a 2-generated bisimple monoid.J. M. Howie [6] proved that any semigroup can be embedded in an idempotentgenerated semigroup. F. Pastijn [9] showed that any semigroup can be embedded in a bisimple idempotent-generated semigroup, and that any countable semigroup can be embedded in a semigroup which is generated by 3 idempotents. Easy proofs of these results using Rees matrix semigroups over a semigroup were given by the author [3]. In this paper, as a corollary to our main result, we deduce that any countable semigroup can be embedded in a bisimple semigroup which is generated by 3 idempotents.

Rees Matrix Covers for Locally Inverse Semigroups

A regular semigroup $S$ is locally inverse if each local submonoid $eSe$, $e$ an idempotent, is an inverse semigroup. It is shown that every locally inverse semigroup is an image of a regular Rees matrix semigroup, over an inverse semigroup, by a homomorphism $\theta$ which is one-to-one on each local submonoid; such a homomorphism is called a local isomorphism. Regular semigroups which are locally isomorphic images of regular Rees matrix semigroups over semilattices are also characterized.

Rees matrix covers for locally inverse semigroups

A regular semigroup S S is locally inverse if each local submonoid e S e eSe , e e an idempotent, is an inverse semigroup. It is shown that every locally inverse semigroup is an image of a regular Rees matrix semigroup, over an inverse semigroup, by a homomorphism θ \theta which is one-to-one on each local submonoid; such a homomorphism is called a local isomorphism. Regular semigroups which are locally isomorphic images of regular Rees matrix semigroups over semilattices are also characterized.

On bisimple semigroups generated by a finite number of idempotents

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.

The structure of pseudo-inverse semigroups

A regular semigroup S is called a pseudo-inverse semigroup if eSe is an inverse semigroup for each e= e2 C S. We show that every pseudo-inverse semigroup divides a semidirect product of a completely simple semigroup and a semilattice. We thereby give a structure theorem for pseudo-inverse semigroups in terms of groups, semilattices and morphisms. The structure theorem which is presented here generalizes several structure theorems which have been given for particular classes of pseudo-inverse semigroups by several authors, and thus contributes to a unification of the theory. Completely (0-) simple semigroups and inverse semigroups form the first prototypes for the study of pseudo-inverse semigroups. We therefore can say that the theory of regular semigroups began with the study of pseudo-inverse semigroups [40, 45]. We may distinguish four successful trends in the papers which since then have dealt with some wider classes of pseudo-inverse semigroups: 1. the subdirect products of completely 0-simple and completely simple semigroups, 2. the generalized inverse semigroups (orthodox pseudo-inverse semigroups, 3. the normal band compositions of inverse semigroups, and 4. Rees matrix semigroups over inverse semigroups (with zero). Subdirect products of completely 0-simple semigroups and completely simple semigroups were initiated in [13, Chapter 2] and studied in great detail in [18] (see also ?4 of [14]); this class contains several interesting subclasses: (a) the trees of completely 0-simple semigroups [18] which include the primitive regular semigroups [7, Vol. II, 16, 39, 44, 46], (b) the regular locally testable semigroups [50] which include the normal bands [36] and the combinatorial completely 0-simple semigroups, (c) the normal bands of groups [37] which include the semilattices of groups [7, Vol. I], (d) the subdirect products of Brandt semigroups which include the locally testable semigroups which are inverse semigroups [50] and the primitive inverse semigroups [39]. The generalized inverse semigroups were introduced in [48] as a special class of orthodox semigroups; they include (a) the inverse semigroups, (b) the orthodox completely 0-simple semigroups [9] and the rectangular groups, (c) the Received by the editors October 25, 1979 and, in revised form, March 25, 1981 and June 18, 1981. AMS (MOS) subject classifications (1970). Primary 20M10.

Matrix semigroups closed under scalar translations

Matrix semigroups closed under scalar translations

Regular four-spiral semigroups, idempotent-generated semigroups and the rees construction

A regular semigroup S is called an ℋ-coextension of a regular semigroup T if there exists an idempotent-separating homomorhism from S onto T. J. Meakin [5] has described all regular four-spiral semigroups, i.e. all ℋ-coextensions of the fundamental four-spiral semigroup Sp4 [2], by means of the structure mappings on a regular semigroup. The purpose of this note is to point out that D. Allen's generalization [1] of the Rees theorem allows one to give a short alternative description of all regular four-spiral semigroups and their maximum completely simple homomorphic images in terms of bisimple ω-semigroups (whose structure is known by Reilly's theorem [7]) and Rees matrix semigroups ℳ(S;I;ΛP) over a semigroup S [3]. The notion of a Rees matrix semigroup over a semigroup is also used to embed semigroups in idempotent-generated ones, providing easy proofs for some embedding theorems of F. Pastijn [6].

Certain rees matrix semigroups with universal properties

There are several instances in recent semigroup literature of completely simple semigroups enjoying certain universal properties relative to the class of all completely simple semigroups or a subclass thereof. The most notable is Clifford's [i] and Rasin's [7] construction of the free completely simple semigroup over a set. Eberhart, williams and Kinch [3] introduced an idempotent generated completely simple semigroup with a prescribed cardinality of the sets of Land R-classes which exhibits a certain type of freeness. Pastijn [5] constructed a free object in the category of idempotent generated completely 0-simple semigroups with a fixed partial groupoid of idempotents. In all these cases, except in [3] where free semigroups and relations are used, a Rees matrix representation of the semigroup in question was explicitly constructed. The subject of the present paper is the "free" object of the following type: prescribed are the cardinalities of the Land

The Ring of Quotients of R[S]; R a Commutative Ring and S a REES Matrix Semigroup over a Semigroup

The Ring of Quotients of <i>R</i>[<i>S</i>]; <i>R</i> a Commutative Ring and <i>S</i> a R<scp>EES</scp> Matrix Semigroup over a Semigroup