Expansions of completely simple semigroups

Abstract

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].

Bifree locally inverse semigroups as expansions

“…Let a, b ∈ I . Then b (b ∧ a) a is an inverse of a b in BFCS(X), and it follows by [7,Proposition 1(i)…”

“…We know from the results in [7] that S(BFCS(X)) is a perfect rectangular band I × I of E-unitary inverse monoids M ab , a, b ∈ I , where (u 1 · · · · · u n )ρ belongs to M ab if and only if λu 1 = a and u n ρ = b. Further, the identity element of M ab is (a ∧ b)ρ .…”

“…We refer to some results of [7]. Let C be a completely simple semigroup and let E C be the subsemigroup of C, generated by the set of idempotents E C .…”

“…So, in the context of varieties, G Pr shifts free objects from the variety of groups up to free objects in the variety of inverse semigroups. Motivated by a recent paper due to Kellendonk and Lawson [13], in [7] we constructed a kind of expansion S(C) for a completely simple semigroup C, which generalizes some important properties of G Pr . For example, as a consequence of the definition G Pr is embeddable into a semidirect product of a semilattice by G. Analogously S(C) embeds into a restricted semidirect product of a semilattice by C. The aim of this paper is to show that also the shifting property for free objects has an analogon, even in the theory of e-varieties, which was introduced by Hall [10] and independently by Kadourek and Szendrei [12] to study regular semigroups from a universal algebraic viewpoint.…”

The generalized Szendrei expansion of an ℜ-unipotent semigroup

“…When S is a group, it admits the form S SL presented by Szendrei [17] which has been extended to both the regular and the nonregular cases. In this last case, it was studied for left cancellative monoids by Fountain and Gomes [5] and for unipotent monoids by Fountain, Gomes and Gould [6]; in the rst, for inverse monoids by Lawson, Margolis and Steinberg [12] and for completely simple semigroups by Billhardt [2]. The inverse monoid construction, named generalized prex expansion and denoted (·) Pr , was suitably modied by Gomes [8] to obtain an expansion for weakly left ample semigroups, an even larger class of non-regular semigroups.…”