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Ehresmann theory and partition monoids
- Publication Year :
- 2021
-
Abstract
- 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.
- Subjects :
- Monoid
Pure mathematics
Algebra and Number Theory
Diagram (category theory)
010102 general mathematics
Block (permutation group theory)
Structure (category theory)
01 natural sciences
Mathematics::Category Theory
0103 physical sciences
Homogeneous space
Partition (number theory)
010307 mathematical physics
0101 mathematics
Bijection, injection and surjection
Quotient
Mathematics
Subjects
Details
- Language :
- English
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....6f106da9ea2ffdad3396965bb0a80eed