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Proper Ehresmann semigroups

Part of: Semigroups Special categories

Published online by Cambridge University Press:  22 August 2023

Ganna Kudryavtseva  and
Valdis Laan
Ganna Kudryavtseva
Affiliation:
Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia (ganna.kudryavtseva@fmf.uni-lj.si)
Valdis Laan
Affiliation:
Institute of Mathematics and Statistics, University of Tartu, Tartu, Estonia (valdis.laan@ut.ee)
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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 σ-class, where σ 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.

Keywords

Ehresmann semigrouprestriction semigroupproper semigroupcoverpartial action

MSC classification

Primary: 20M10: General structure theory 20M30: Representation of semigroups; actions of semigroups on sets 18B40: Groupoids, semigroupoids, semigroups, groups (viewed as categories)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 66 , Issue 3 , August 2023 , pp. 758 - 788
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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