Ehresmann monoids
Branco, Mário J. J.; Gomes, Gracinda M. S.; Gould, Victoria
J. Algebra , 443 (2015), 349–382
https://doi.org/10.1016/j.jalgebra.2015.06.035
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
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