Properties of congruences of twisted partition monoids and their lattices
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.
Citation
East , J & Ruskuc , N 2022 , ' Properties of congruences of twisted partition monoids and their lattices ' , Journal of the London Mathematical Society , vol. 106 , no. 1 , pp. 311-357 . https://doi.org/10.1112/jlms.12574
Publication
Journal of the London Mathematical Society
Status
Peer reviewed
ISSN
0024-6107Type
Journal article
Description
Funding: The first author is supported by ARC Future Fellowship FT190100632. The second author is supported by EPSRC grant EP/S020616/1.Collections
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