Ehresmann monoids

Ehresmann monoids form a variety of bi-unary monoids, that is, monoids equipped with two basic unary operations, the images of which coincide and form a semilattice of projections  . The monoid of binary relations BXBX on any set X   with unary operations of domain and range is Ehresmann. Inverse monoids, regarded as bi-unary submonoids of BXBX via the Wagner–Preston representation theorem, are therefore also Ehresmann. At the other extreme, any monoid is Ehresmann, where the unary operations take all elements to the monoid identity. We demonstrate here using semilattices and monoids as building blocks that Ehresmann monoids have a rich structure, fundamentally different from that of inverse monoids and, indeed, from that of the interim class of restriction monoids.The article introduces a notion of properness   for Ehresmann monoids, that tightly controls structure and is dependent upon sets of generators. We show how to construct an Ehresmann monoid P(T,Y)P(T,Y) satisfying our properness condition from a semilattice Y acted upon on both sides by a monoid T via order preserving maps. The free Ehresmann monoid on X   is proven to be of the form P(X⁎,Y)P(X⁎,Y). The next question deals with the existence of proper covers. We answer it in a positive way, proving that any Ehresmann monoid M   admits a cover of the form P(X⁎,E)P(X⁎,E), where E is the semilattice of projections of M. Here a ‘cover’ is a preimage under a morphism that separates elements in E.