Ehresmann monoids: adequacy and expansions

Journal of Algebra, 513 (2018), 344-367
https://doi.org/10.1016/j.jalgebra.2018.06.036

It is known that an Ehresmann monoid may be constructed from a monoid T acting via order-preserving maps on both sides of a semilattice Y with identity, such that the actions satisfy an appropriate compatibility criterion. Our main result shows that if T is cancellative and equidivisible (as is the case for the free monoid ?), the monoid

not only is Ehresmann but also satisfies the stronger property of being adequate.

Fixing T, Y and the actions, we characterise
as being unique in the sense that it is the initial object in a suitable category of Ehresmann monoids. We also prove that the operator defines an expansion of Ehresmann monoids.