Expansions of completely simple semigroups

For any completely simple semigroup C a regular expansion S(C) is constructed which is the Birget-Rhodes prefix expansion CPr if C is a group [6]. We show that our construction generalizes two important features of CPr. Moreover we embed S (C) into a restricted semidirect product of a semilattice by C and investigate the relationship to the expansion P(C), introduced by Meakin [14].</jats:p

A factorization of dual prehomomorphisms and expansions of inverse semigroups

For any inverse semigroup S we construct an inverse semigroup S(S), which has the following universal property with respect to dual prehomomorphisms from S: there is an injective dual prehomomorphism ιS: S → S(S) such that for each dual prehomomorphism θ from S into an inverse semigroup T there exists a unique homomorphism θ* : S(S) → T with ιSθ* = θ. If we restrict the class of dual prehomomorphisms under consideration to order preserving ones, S(S) may be replaced by a certain homomorphic image Ŝ(S) which can be viewed as a natural generalization of the Birget-Rhodes prefix expansion for groups [4] to inverse semigroups. Recently, Lawson, Margolis and Steinberg [8] have given an alternative description of Ŝ(S) which is based on O’Carroll’s theory of idempotent pure congruences [11]. It should be noted that our ideas can be used to simplify some of their arguments.</jats:p

Extensions of left regular bands by R-unipotent semigroups

In this paper we describe R-unipotent semigroups being regular extensions of a left regular band by an R-unipotent semigroup T as certain subsemigroups of a wreath product of a left regular band by T .We obtain Szendrei’s result that each E-unitaryR-unipotent semigroup is embeddable into a semidirect product of a left regular band by a group. Further, specialising the first author’s notion of λ-semidirect product of a semigroup by a locally R-unipotent semigroup, we provide an answer to an open question raised by the authors in [Extensions and covers for semigroups whose idempotents form a left regular band, Semigroup Forum 81 (2010), 51-70]

Weakly E-unitary locally inverse semigroups

AbstractWe prove that each weakly E-unitary locally inverse semigroup is embeddable in a restricted semidirect product of a normal band by a completely simple semigroup and, equivalently, in a Pastijn product of a normal band by a completely simple semigroup

A class of unary semigroups admitting a rees matrix representation

In [1] the authors introduced the notion of an associate inverse subsemigroup of a regular semigroup, extending the concept of an associate subgroup of a regular semigroup, ¯rst presented in [3]. The main result of the present paper, Theorem 2.15, establishes that a regular semigroup S with an associate inverse subsemigroup S* satisfies three simple identities if and only if it is isomorphic to a generalised Rees matrix semigroup M(T; A;B; P), where T is a Clifford semigroup, A and B are bands, with common associate inverse subsemigroup E(T) satisfying the referred identities, and P is a sandwich matrix satisfying some natural conditions. If T is a group and A, B are left and right zero semigroups, respectively, then the structure described provides a usual Rees matrix semigroup with normalised sandwich matrix, thus generalising the Rees matrix representation for completely simple semigroups.This research was financed by FEDER Funds through "Programa Operacional Factores de Competitividade "COMPETE" and by Portuguese Funds through FCT-"Fundacao para a Ciencia e a Tecnologia", within the project PEst-C/MAT/UI0013/2011.This research was also financed by the Portuguese Government through the FCT Fundacao para a Ciencia e a Tecnologia under the project PEst-OE/MAT/UI4080/2011