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Abstract
We build on the recent characterisation of congruences on the infinite twisted partition monoids PΦn and their finite d-twisted homomorphic images PΦn,d, and investigate their algebraic and order-theoretic properties. We prove that each congruence of PΦn is (finitely) generated by at most ⌈5n/2⌉ pairs, and we characterise the principal ones. We also prove that the congruence lattice Cong(PΦn) is not modular (or distributive); it has no infinite ascending chains, but it does have infinite descending chains and infinite antichains. By way of contrast, the lattice Cong(PΦn,d) is modular but still not distributive for d>0, while Cong(PΦn,0) is distributive. We also calculate the number of congruences of PΦn,d, showing that the array (|Cong(PΦn,d)|)n,d≥0 has a rational generating function, and that for a fixed n or d, |Cong(PΦn,d)| is a polynomial in d or n≥4, respectively.
Original language | English |
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Pages (from-to) | 311-357 |
Journal | Journal of the London Mathematical Society |
Volume | 106 |
Issue number | 1 |
Early online date | 15 Mar 2022 |
DOIs | |
Publication status | Published - Jul 2022 |
Keywords
- Partition monoid
- Twisted partition monoid
- Congruence
- Finitely generated congruence
- Principal congruence
- Congruence lattice
- Modular lattice
- Distributive lattice
- Enumeration
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Dive into the research topics of 'Properties of congruences of twisted partition monoids and their lattices'. Together they form a unique fingerprint.Projects
- 1 Finished
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Diagram Monoids and Their Congruences: Diagram Monoids and Their Congruences
Ruskuc, N. (PI)
15/12/18 → 14/02/21
Project: Standard