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 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 that form. 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 that form, where E is the semilattice of projections of M. Here a ‘cover’ is a preimage under a morphism that separates elements in E

CEMAT - Center for Computational and Stochastic Mathematics