Ehresmann monoids: adequacy and expansions
Branco, Mário J. J.; Gomes, Gracinda M. S.; Gould, Victoria; Wang, Yanhui
Journal of Algebra, 513 (2018), 344-367
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.