Left Ehresmann monoids with a proper basis
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| Formato: | Preprint |
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2026
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| _version_ | 1866918467171516416 |
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| author | Gomes, Gracinda Gould, Victoria Wang, Yanhui |
| author_facet | Gomes, Gracinda Gould, Victoria Wang, Yanhui |
| contents | Left Ehresmann monoids, and their two-sided counterpart of Ehresmann monoids, were so named by Lawson, who elucidated their connection to the work of Ehresmann in differential geometry. This article is dedicated to building a theory for left Ehresmann monoids inspired by that for inverse semigroups; in order to do so we must develop substantially different ideas and techniques. 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 notion of a proper basis, and show that $\mathcal{P}_{\ell}(T,X)$, and consequently any free left Ehresmann monoid, possesses a proper basis. We show that any left Ehresmann monoid with a proper basis displays properties close to those of two-sided Ehresmann monoids. 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. We prove that any left Ehresmann monoid with a proper basis is isomorphic to some $\mathcal{Q}_{\ell}(T,X,Y)$. 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.
In the process of proving our main theorems we present a globalisation result for an order-preserving partial action of a monoid on a partially ordered set or semilattice. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_22923 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Left Ehresmann monoids with a proper basis Gomes, Gracinda Gould, Victoria Wang, Yanhui Rings and Algebras 20M10, 20M30 Left Ehresmann monoids, and their two-sided counterpart of Ehresmann monoids, were so named by Lawson, who elucidated their connection to the work of Ehresmann in differential geometry. This article is dedicated to building a theory for left Ehresmann monoids inspired by that for inverse semigroups; in order to do so we must develop substantially different ideas and techniques. 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 notion of a proper basis, and show that $\mathcal{P}_{\ell}(T,X)$, and consequently any free left Ehresmann monoid, possesses a proper basis. We show that any left Ehresmann monoid with a proper basis displays properties close to those of two-sided Ehresmann monoids. 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. We prove that any left Ehresmann monoid with a proper basis is isomorphic to some $\mathcal{Q}_{\ell}(T,X,Y)$. 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. In the process of proving our main theorems we present a globalisation result for an order-preserving partial action of a monoid on a partially ordered set or semilattice. |
| title | Left Ehresmann monoids with a proper basis |
| topic | Rings and Algebras 20M10, 20M30 |
| url | https://arxiv.org/abs/2601.22923 |