Ehresmann theory and partition monoids

East, James and Gray, Robert (2021) Ehresmann theory and partition monoids. Journal of Algebra, 579. pp. 318-352. ISSN 0021-8693

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This article concerns Ehresmann structures in the partition monoid P X. Since P X contains the symmetric and dual symmetric inverse monoids on the same base set X, it naturally contains the semilattices of idempotents of both submonoids. We show that one of these semilattices leads to an Ehresmann structure on P X while the other does not. We explore some consequences of this (structural/combinatorial and representation theoretic), and in particular characterise the largest left-, right- and two-sided restriction submonoids. The new results are contrasted with known results concerning relation monoids, and a number of interesting dualities arise, stemming from the traditional philosophies of inverse semigroups as models of partial symmetries (Vagner and Preston) or block symmetries (FitzGerald and Leech): “surjections between subsets” for relations become “injections between quotients” for partitions. We also consider some related diagram monoids, including rook partition monoids, and state several open problems.

Item Type: Article
Uncontrolled Keywords: dual symmetric inverse monoids,ehresmann categories,ehresmann monoids,partition monoids,relation monoids,symmetric inverse monoids,algebra and number theory ,/dk/atira/pure/subjectarea/asjc/2600/2602
Faculty \ School: Faculty of Science > School of Mathematics
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Depositing User: LivePure Connector
Date Deposited: 29 Mar 2021 23:56
Last Modified: 23 Oct 2022 02:20
DOI: 10.1016/j.jalgebra.2021.02.038

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