Almost perfect restriction semigroups

Abstract

Abstract:We call a restriction semigroup almost perfect if it is proper and the least congruence that identifies all its projections 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 TIY 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.

Partial monoid actions and a class of restriction semigroups

“…To show this, we construct a projection separating (2, 1, 1)-congruence κ on W (T, X), which extends the congruence on M(T, Y ) mapping it onto S, so that S embeds into W (T, X)/κ. We conclude the introduction by pointing out that, after an early version of this paper existed, the author learned that ultra proper restriction semigroups and ultra F -restriction monoids were independently introduced and studied (from a somewhat different perspective) by Peter Jones in [12] under the names almost perfect restriction semigroups and perfect restriction monoids, respectively. This terminology is motivated by an elegant characterization noticed in [12] that a proper restriction semigroup is ultra proper if and only if the least congruence identifying all projections is perfect meaning that the product of classes is again a whole class.…”

Embedding in factorisable restriction monoids

“…In the last section, we strengthen some results in [8] and [7] by proving that each ultra F -restriction, or, equivalently, each perfect restriction monoid S whose greatest reduced factor is a free monoid is embeddable in a semidirect product of a semilattice by a monoid in such a way that the monoid acts on the semilattice by automorphisms, and all congruences of S extend to the semidirect product.…”

“…Earlier works [7], [8], [11], [12] have achieved embeddings of some or all restriction semigroups in members of wider classes that are inherently one-sided. The difficulty of our task is perhaps understood when we remark that it is undecidable whether a finite restriction semigroup embeds as a restriction semigroup into an inverse semigroup [6].…”

Two-sided expansions of monoids

“…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 .…”