Partial monoid actions and a class of restriction semigroups

Abstract

We study classes of proper restriction semigroups determined by properties of partial actions underlying them. These properties include strongness, antistrongness, being defined by a homomorphism, being an action etc. Of particular interest is the class determined by homomorphisms, primarily because we observe that its elements, while being close to semidirect products, serve as mediators between general restriction semigroups and semidirect products or W -products in an embedding-covering construction. It is remarkable that this class does not have an adequate analogue if specialized to inverse semigroups. F -restriction monoids of this class, called ultra F -restriction monoids, are determined by homomorphisms from a monoid T to the Munn monoid of a semilattice Y . We show that these are precisely the monoids Y * m T considered by Fountain, Gomes and Gould. We obtain a McAlister-type presentation for the class given by strong dual prehomomorphisms and apply it to construct an embedding of ultra F -restriction monoids, for which the base monoid T is free, into W -products of semilattices by monoids. Our approach yields new and simpler proofs of two recent embedding-covering results by Szendrei.

Two-sided expansions of monoids

“…In this subsection we recall the Cornock-Gould structure result on proper restriction monoids [7]. We remark that in [7] a pair of partial actions, called a double action, satisfying certain compatibility conditions, was considered, and in [23] we reformulated this using one partial action by partial bijections.…”

“…Here we restate the construction of [23] in terms of premorphisms. Let T be a monoid, Y a semilattice with top element e and assume that we are given a premorphism ϕ : T → I(Y ).…”

“…A restriction semigroup is called an F -restriction semigroup if every its σ-class has a maximum element. It is easy to check (or see [23,Lemma 5]) that an F -restriction semigroup is necessarily a monoid with the identity element being the maximum projection. An F -restriction monoid is necessarily proper.…”

“…It is called left extra F -restriction (resp. right extra F -restriction or extra F -restriction) [23] provided that the underling premorphism ϕ is left strong (resp. right strong or strong).…”

“…right strong or strong). S is called ultra F -restriction [23] (or perfect [21]) if ϕ is a homomorphism. Proposition 4.2 tells us that all these classes can be equivalently defined by the respective properties of the premorphism τ .…”