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arXiv · 2203.07129

Proper Ehresmann semigroups

Ganna Kudryavtseva·Valdis Laan
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Abstract

We propose a notion of a proper Ehresmann semigroup based on a three-coordinate description of its generating elements governed by certain labelled directed graphs with additional structure. The generating elements are determined by their domain projection, range projection and $\sigma$-class, where $\sigma$ denotes the minimum congruence that identifies all projections. We prove a structure result on proper Ehresmann semigroups and show that every Ehresmann semigroup has a proper cover. Our covering monoid turns out to be isomorphic to that from the work by Branco, Gomes and Gould and provides a new view of the latter. Proper Ehresmann semigroups all of whose elements admit a three-coordinate description are characterized in terms of partial multiactions of monoids on semilattices. As a consequence we recover the two-coordinate structure result on proper restriction semigroups.

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Ganna Kudryavtseva, Valdis Laan. 2022-03-14. Proper Ehresmann semigroups. https://doi.org/10.1017/s0013091523000391

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