Rees Matrix Research Articles

Rees Matrix Covers and the Translational Hull of a Locally Inverse Semigroup

For any locally inverse semigroup S, there exists a maximal dense ideal extension of S within the class LI of all locally inverse semigroups (Pastijn and Oliveira, 2006, Preprint). Here we realize this maximal dense ideal extension in terms of a canonically constructed quotient of a regular Rees matrix semigroup over an inverse semigroup.

REES MATRIX THEOREM FOR $\mathcal{D}^{(\ell)}$-SIMPLE STRONGLY RPP SEMIGROUPS

A semigroup S is called rpp if all right principal ideals of S, regarded as S1-systems, are projective. An rpp semigroup S is said to be strongly rpp if for any a ∈ S, there exists a unique idempotent e such that [Formula: see text] and a = ea. In this paper, we show that a [Formula: see text]-simple strongly rpp semigroup can be expressed by a Rees matrix semigroup over a left cancellative monoid and conversely. Our result generalizes the classical theorem of Rees in 1940 and also amplifies the Rees theorem in semigroup given by Lallement and Petrich in 1969.

Automatic structures for semigroup constructions

We survey results concerning automatic structures for semigroup constructions, providing references and describing the corresponding automatic structures. The constructions we consider are: free products, direct products, Rees matrix semigroups, Bruck-Reilly extensions and wreath products.

The loop problem for Rees matrix semigroups

We study the relationship between the loop problem of a semigroup, and that of a Rees matrix construction (with or without zero) over the semigroup. This allows us to characterize exactly those completely zero-simple semigroups for which the loop problem is context-free. We also establish some results concerning loop problems for subsemigroups and Rees quotients.

Characterizing cryptogroups with a finite number of $${\mathcal{H}}$$ -classes in each $${\mathcal{D}}$$ -class by their subsemigroups

We construct a family \({\mathcal{F}}\) of completely regular semigroups with the property that each completely regular semigroup S with a finite number of \({\mathcal{H}}\) -classes in each \({\mathcal{D}}\) -class is non-cryptic if and only if S contains an isomorphic image of a member of \({\mathcal{F}}\) . Each member F of \({\mathcal{F}}\) is an ideal extension of a Rees matrix semigroup J by a cyclic group B with a zero adjoined and the identity of B is the identity of F. Here \(J={\mathcal{M}}\left( I,G,\Lambda;P\right) \) with I and Λ finite, G is given by generators and relations, and P is given explicitly. Within completely regular semigroups, the cryptic property is equivalent to \({\mathcal{N}=\mathcal{S}},\) where \({\mathcal{N}}\) is the natural partial order and a\({\mathcal{S}} b\) if and only if a2 = ab = ba. Hence the above result can be formulated in terms of \({\mathcal{N}}\) and \({\mathcal{S}}\) .

Automatic Rees Matrix Semigroups over Categories

We consider the preservation of the properties of automaticity and prefix-automaticity in Rees matrix semigroups over semigroupoids and small categories. Some of our results are new or improve upon existing results in the single-object case of Rees matrix semigroups over semigroups.

Derived Rees Matrix Semigroups as Semigroups of Transformations

An ordered pair (e,f) of idempotents of a regular semigroup is called a skew pair if ef is not idempotent whereas fe is idempotent. Previously [1] we have established that there are four distinct types of skew pairs of idempotents. We have also described (as quotient semigroups of certain regular Rees matrix semigroups [2]) the structure of the smallest regular semigroups that contain precisely one skew pair of each of the four types, there being to within isomorphism ten such semigroups. These we call the derived Rees matrix semigroups. In the particular case of full transformation semigroups we proved in [3] that TX contains all four skew pairs of idempotents if and only if |X| ≥ 6. Here we prove that TX contains all ten derived Rees matrix semigroups if and only if |X| ≥ 7.

Regular Semigroups Each of Whose Least Completely Simple Congruence Classes Has a Greatest Element

A regular semigroup S is termed locally F-regular, if each class of the least completely simple congruence ξ contains a greatest element with respect to the natural partial order. It is shown that each locally F-regular semigroup S admits an embedding into a semidirect product of a band by S/ξ. Further, if ξ satisfies the additional property that for each s ∈ S and each inverse (sξ)′ of sξ in S/ξ the set (sξ)′ ∩ V(s) is not empty, we represent S both as a Rees matrix semigroup over an F-regular semigroup as well as a certain subsemigroup of a restricted semidirect product of a band by S/ξ.

Finite derivation type for Rees matrix semigroups

This paper introduces the topological finiteness condition finite derivation type (FDT) on the class of semigroups. This notion is naturally extended from the monoid case. With this new concept we are able to prove that if a Rees matrix semigroup M [ S ; I , J ; P ] has FDT then the semigroup S also has FDT. Given a monoid S and a finitely presented Rees matrix semigroup M [ S ; I , J ; P ] we prove that if the ideal of S generated by the entries of P has FDT, then so does M [ S ; I , J ; P ] . In particular, we show that, for a finitely presented completely simple semigroup M , the Rees matrix semigroup M = M [ S ; I , J ; P ] has FDT if and only if the group S has FDT.

Uniform decision problems for automatic semigroups

We consider various decision problems for automatic semigroups, which involve the provision of an automatic structure as part of the problem instance. With mild restrictions on the automatic structure, which are necessary to make the problem well defined, the uniform word problem for semigroups described by automatic structures is decidable. Under the same conditions, we show that one can also decide whether the semigroup is completely simple or completely zero-simple; in the case that it is, one can compute a Rees matrix representation for the semigroup, in the form of a Rees matrix together with an automatic structure for its maximal subgroup. On the other hand, we show that it is undecidable in general whether a given element of a given automatic monoid has a right inverse.

Generating Sets of Completely 0-Simple Semigroups

ABSTRACT A formula for the rank of an arbitrary finite completely 0-simple semigroup, represented as a Rees matrix semigroup ℳ0[G; I, Λ; P], is given. The result generalizes that of Ruškuc concerning the rank of connected finite completely 0-simple semigroups. The rank is expressed in terms of |I|, |Λ|, the number of connected components k of P, and a number r min, which we define. We go on to show that the number r min is expressible in terms of a family of subgroups of G, the members of which are in one-to-one correspondence with, and determined by the nonzero entries of, the components of P. A number of applications are given, including a generalization of a result of Gomes and Howie concerning the rank of an arbitrary Brandt semigroup B(G,{1,…,n}).

Rees matrix semigroups and the regular semidirect product

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.

FAITHFUL FUNCTORS FROM CANCELLATIVE CATEGORIES TO CANCELLATIVE MONOIDS WITH AN APPLICATION TO ABUNDANT SEMIGROUPS

We prove that any small cancellative category admits a faithful functor to a cancellative monoid. We use our result to show that any primitive ample semigroup is a full subsemigroup of a Rees matrix semigroup [Formula: see text] where M is a cancellative monoid and P is the identity matrix. On the other hand a consequence of a recent result of Steinberg is that it is undecidable whether a finite ample semigroup embeds as a full subsemigroup of an inverse semigroup.

On finiteness conditions for Rees matrix semigroups

Let T=ℳ[S; I; J; P] be a Rees matrix semigroup where S is a semigroup, I and J are index sets, and P is a J × I matrix with entries from S, and let U be the ideal generated by all the entries of P. If U has finite index in S, then we prove that T is periodic (locally finite) if and only if S is periodic (locally finite). Moreover, residual finiteness and having solvable word problem are investigated.

PRESENTATIONS FOR SEMIGROUPS AND SEMIGROUPOIDS

We consider the relationship between the combinatorial properties of semigroupoids in general and semigroups in particular. We show that a semigroupoid is finitely generated [finitely presentable] exactly if the corresponding categorical-at-zero semigroup is finitely generated [respectively, finitely presentable]. This allows us to extend some of the main results of [17], to show that finite generation and presentability are preserved under finite extension of semigroupoids and the taking of cofinite subsemigroupoids. We apply this result to extend the results of [6], giving characterizations of finite generation and finite presentability in Rees matrix semigroups over semigroupoids.

THE PERKINS SEMIGROUP HAS CO-NP-COMPLETE TERM-EQUIVALENCE PROBLEM

A semigroup term is a finite word in the alphabet x1, x2,…. The length of a term p, denoted by |p|, is the number of variables in p, including multiplicities. The term-equivalence problem for a finite semigroup S has as an instance a pair of terms {p,q} with size |p| + |q| and asks whether p ≈ q is an identity over S. It is proved here that [Formula: see text], the six-element Perkins semigroup, has co-NP-complete term-equivalence problem, a result which leads to the completion of the classification of he term-equivalence problems for monoid extensions of aperiodic Rees matrix semigroups. From the main result it follows that there exist finite semigroups with tractable term-equivalence problems but having subsemigroups and homomorphic images with co-NP-complete term-equivalence problems.

On Classes of E-Inversive Semigroups and Semigroups Whose Idempotents Form a Subsemigroup#

A semigroup S is called E-inversive if for every a ∈ S there is an x ∈ S such that ax is idempotent. The purpose of this paper is the investigation of E-inversive semigroups and semigroups whose idempotents form a subsemigroup. Basic properties are analysed and, in particular, semigroups whose idempotents form a semilattice or a rectangular band are considered. To provide examples and characterizations, the construction methods of generalized Rees matrix semigroups and semidirect products are employed.

On Locally Inverse Semigroups Having Weakly E-unitary Covers

Recently, Billhardt has characterized the locally inverse semigroups embeddable in Rees matrix semigroups over generalized inverse semigroups. We prove here that these semigroups are just the locally inverse semigroups having weakly E-unitary covers.

The Smallest Regular Semigroups with all Four Skew Pairs of Idempotents

An ordered pair (e, f) of idempotents of a regular semigroup is called a skew pair if ef is not idempotent whereas fe is idempotent. We have shown previously that there are four distinct types of skew pairs of idempotents. Here we consider the smallest regular semigroups that contain precisely one of each of these four types. We show that, to within isomorphism and dualisomorphism, there are six such semigroups and characterise them as quotient semigroups of certain regular Rees matrix semigroups.

Embedding Semigroups into Idempotent Generated Ones

We present a concrete model of the embedding due to Pastijn and Yan of a semigroup S into an idempotent generated semigroup now in terms of a Rees matrix semigroup over S1. The paper starts with a comparison of the two embeddings. Studying the properties of this embedding, we prove that it is functorial. We show that a number of usual semigroup properties is preserved by this embedding, such as periodicity, finiteness, the cryptic property, regularity, complete semisimplicity and various local properties, but complete regularity is not one of them.