The Free Ample Monoid

Abstract

We show that the free weakly E-ample monoid on a set X is a full submonoid of the free inverse monoid FIM(X) on X. Consequently, it is ample, and so coincides with both the free weakly ample and the free ample monoid FAM(X) on X. We introduce the notion of a semidirect product Y*T of a monoid T acting doubly on a semilattice Y with identity. We argue that the free monoid X* acts doubly on the semilattice [Formula: see text] of idempotents of FIM(X) and that FAM(X) is embedded in [Formula: see text]. Finally we show that every weakly E-ample monoid has a proper ample cover.

Algebraic properties of Zappa–Szép products of semigroups and monoids

“…We note that Ker β intersects trivially with R E . But we cannot expect T to be proper in general as T contains an isomorphic copy of S. With a different approach, retaining the same action of S on E, but E acting trivially on S, we obtain a semidirect product E S and this approach yields a proper cover of any left restriction semigroup as proved by Manes [16] (see also [5,7]). Manes' left restriction subsemigroup of E S is given as follows: P = {(e, s) : e ≤ s + } which becomes a proper cover for S under (e, s) → es.…”

“…For (left) restriction semigroups, many authors have produced work using constructions similar (at least on the surface) to those of McAlister, replacing groups by monoids of various kinds. Covering results for restriction semigroups are known due to Cornock, Fountain, Gomes and Gould [3,5,7,8]. The structure of proper restriction semigroups is determined by analogues of P-semigroups.…”

“…The same is true for the semidirect product of a left restriction semigroup and a semilattice. But in the special case where S is a monoid (because monoids are left restriction), it is proved by Fountain et al [5] that the semidirect product of a monoid with a semilattice is again left restriction. A similar result was proved by Manes [16] in the guise of guarded semigroups and by Gomes and Gould [7] in the case of unipotent monoids for weakly left ample semigroups, where a weakly left ample semigroup is a left restriction semigroup with E = E(S).…”

“…To overcome this, we use the notion of a double action (introduced in [5]) and compatibility conditions (introduced in [3]). …”

Retracts of trees and free left adequate semigroups

“…The generalization of our results to categories thus relates to our main results in the same way that the description of the free restriction category on a graph given in [2] relates to the descriptions of free left ample monoids given by Fountain et al . [6,7].…”