Structure of Left Adequate and Left Ehresmann Monoids

This is the first of two articles studying the structure of left adequate and, more generally, of left Ehresmann monoids. Motivated by a careful analysis of normal forms, we introduce here a concept of proper for a left adequate monoid M . In fact, our notion is that of T -proper, where T is a submonoid of M . We show that any left adequate monoid M has an X∗proper cover for some set X , that is, there is a left adequate monoid M̂ that is X∗-proper, and an idempotent separating epimorphism θ : M̂ → M of the appropriate type. Given this result, we may deduce that the free left adequate monoid on any set X is X∗-proper. In a subsequent paper, we show how to construct T -proper left adequate monoids from any monoid T acting via order preserving maps on a semilattice with identity, and prove that the free left adequate monoid is of this form. 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. We also indicate how to obtain some of the analogous results in the two-sided case. This paper and its sequel, 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.