Almost perfect restriction semigroups

TL;DR

This paper characterizes almost perfect restriction semigroups as specific W-products involving monoids and semilattices, establishing a McAlister-type theorem and emphasizing their rich structure beyond inverse semigroup theory.

Contribution

It introduces a new characterization of almost perfect restriction semigroups via W-products and proves a McAlister-type theorem for all restriction semigroups.

Findings

Almost perfect restriction semigroups are isomorphic to W-products W(T,Y).

Every restriction semigroup has an easily computed cover of the W-product type.

Main results generalize Szendrei's description and include classes of free restriction semigroups.

Abstract

We call a restriction semigroup almost perfect if it is proper and its least monoid congruence is perfect. We show that any such semigroup is isomorphic to a `$W$-product' $W(T,Y)$, where $T$ is a monoid, $Y$ is a semilattice and there is a homomorphism from $T$ into the inverse semigroup $TI_Y$ of isomorphisms between ideals of $Y$. Conversely, all such $W$-products are almost perfect. Since we also show that every restriction semigroup has an easily computed cover of this type, the combination yields a `McAlister-type' theorem for all restriction semigroups. It is one of the theses of this work that almost perfection and perfection, the analogue of this definition for restriction monoids, are the appropriate settings for such a theorem. That these theorems do not reduce to a general theorem for inverse semigroups illustrates a second thesis of this work: that restriction (and, by extension, Ehresmann) semigroups have a rich theory that does not consist merely of generalizations of inverse semigroup theory. It is then with some ambivalence that we show that all the main results of this work easily generalize to encompass all proper restriction semigroups. The notation $W(T,Y)$ recognizes that it is a far-reaching generalization of a long-known similarly titled construction. As a result, our work generalizes Szendrei's description of almost factorizable semigroups while at the same time including certain classes of free restriction semigroups in its realm.