Restriction semigroups and λ -Zappa-Szép products

The aim of this paper is to study λ-semidirect and λ-Zappa-Szép products of restriction semigroups. The former concept was introduced for inverse semigroups by Billhardt, and has been extended to some classes of left restriction semigroups. The latter was introduced, again in the inverse case, by Gilbert and Wazzan. We unify these concepts by considering what we name the scaffold of a Zappa-Szép product S⋈ T where S and T are restriction. Under certain conditions this scaffold becomes a category. If one action is trivial, or if S is a semilattice and T a monoid, the scaffold may be ordered so that it becomes an inductive category. A standard technique, developed by Lawson and based on the Ehresmann-Schein-Nambooripad result for inverse semigroups, allows us to define a product on our category. We thus obtain restriction semigroups that are λ-semidirect products and λ-Zappa-Szép products, extending the work of Billhardt and of Gilbert and Wazzan. Finally, we explicate the internal structure of λ-semidirect products

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]

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

The variety of unary semigroups with associate inverse subsemigroup

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, first presented in [2]. The semigroups of these two classes admit axiomatic characterisations in terms of unary operations and can, therefore, be considered as unary semigroups. In this paper we introduce the notion of unary semigroup with associate inverse subsemigroup [with associate subgroup] and show that the classes of such unary semigroups form varieties.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