Left Adequate and Left Ehresmann Monoids Ii

This article is the second of two presenting a new approach to left adequate monoids. In the first, we introduced the notion of being T -proper, where T is a submonoid of a left adequate monoid M . We showed that the free left adequate monoid on a set X is X∗-proper. Further, any left adequate monoid M has an X∗-proper cover for some set X , that is, there is an X∗proper left adequate monoid M̂ and an idempotent separating epimorphism θ : M̂ → M of the appropriate signature. We now show how to construct a T -proper left adequate monoid P(T, Y ) from a monoid T acting via order preserving maps on a semilattice Y with identity. Our construction plays the role for left adequate monoids that the semidirect product of a group and a semilattice plays for inverse monoids. A left adequate monoid M with semilattice E has an X∗-proper cover P(X∗, E). Hence, by choosing a suitable semilattice EX and an action of X ∗ on EX , we prove that the free left adequate monoid is of the form P(X∗, EX). An alternative description of the free left adequate monoid appears in a recent preprint of Kambites. We show how to obtain the labelled trees appearing in his result from our structure theorem. Our results apply to the wider class of left Ehresmann monoids, and we give them in full generality. Indeed this is the right setting: the class of left Ehresmann monoids is the variety generated by the quasi-variety of left adequate monoids. This paper, and the two of Kambites on free (left) adequate semigroups, demonstrate the rich but accessible structure of (left) adequate semigroups and monoids, introduced with startling insight by Fountain some 30 years ago.