Left Ehresmann monoids with a proper basis
Gracinda Gomes, Victoria Gould, Yanhui Wang
TL;DR
This work extends the McAlister–O'Carroll paradigm to left Ehresmann monoids by introducing the $*$-left Ehresmann framework and the notion of a proper basis. It constructs the canonical left-Ehresmann objects $\mathcal{P}_{\ell}(T,X)$ and their subsemigroups $\mathcal{Q}_{\ell}(T,X,Y)$ via order-preserving partial actions and their globalisations, proving that every such monoid with a proper basis is isomorphic to a $\mathcal{Q}_{\ell}(T,\mathcal{X},\mathcal{Y})$ and that free left Ehresmann monoids fit into this picture. The main structural theorem provides a precise classification, showing $Q$ is a biunary monoid with a canonical decomposition iff it arises from a $\mathcal{Q}_{\ell}$-construction inside a suitable $\mathcal{P}_{\ell}$-type semigroup. By leveraging $T$-normal forms, canonical bases, and globalised partial actions, the paper unifies left Ehresmann theory with semidirect-product-like decompositions and offers a robust platform for further development of left-restriction/ample-type structures. Overall, this yields a comprehensive, classification-oriented framework for left Ehresmann monoids with proper bases, paralleling the inverse-semigroup theory.
Abstract
This article gives an abstract characterisation of a class of left Ehresmann monoids possessing certain universal properties. It is known that every left Ehresmann monoid has a cover, that is, a projection separating preimage, of the form $\mathcal{P}_{\ell}(T,X)$, where $\mathcal{P}_{\ell}(T,X)$ is a left Ehresmann monoid constructed from a monoid $T$ and an order-preserving action of $T$ on a semilattice $X$ with identity. We introduce the class of $*$-left Ehresmann monoids and show that each $\mathcal{P}_{\ell}(T,X)$ belongs to this class; in particular so does any free left Ehresmann monoid. Further, we present the notion of a proper basis, and show that $\mathcal{P}_{\ell}(T,X)$ possesses a proper basis. Next, we exhibit a class of subsemigroups $\mathcal{Q}_{\ell}(T,X,Y)$ (properly, biunary monoid subsemigroups) of the monoids $\mathcal{P}_{\ell}(T,X)$ which are also $*$-left Ehresmann with a proper basis, and prove that up to isomorphism they form exactly the class of all such monoids. Our results can be regarded as being analogous to those for proper inverse semigroups, due to McAlister and O'Carroll, the $\mathcal{Q}_{\ell}(T,X,Y)$ playing the role of the $P$-semigroups and the $\mathcal{P}_{\ell}(T,X)$ the role of the semidirect products of a semilattice by a group.