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
Rights
Copyright © 2022 The Authors. Journal of the London Mathematical Society is copyright © London Mathematical Society. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
Description
Funding: The first author is supported by ARC Future Fellowship FT190100632. The second author is supported by EPSRC grant EP/S020616/1.Collections
Items in the St Andrews Research Repository are protected by copyright, with all rights reserved, unless otherwise indicated.
Related items
Showing items related by title, author, creator and subject.
-
The lattice of one-sided congruences on an inverse semigroup
Brookes, Matthew (2023-09-01) - Journal articleWe build on the description of left congruences on an inverse semigroup in terms of the kernel and trace due to Petrich and Rankin. The notion of an inverse kernel for a left congruence is developed. Various properties of ... -
Classification of congruences of twisted partition monoids
East, James; Ruskuc, Nik (2021-11-18) - Journal articleThe twisted partition monoid PΦn is an infinite monoid obtained from the classical finite partition monoid Pn by taking into account the number of floating components when multiplying partitions. The main result of this ... -
Congruence lattices of ideals in categories and (partial) semigroups
East, James; Ruskuc, Nik (2020-05-25) - Journal articleThis paper presents a unified framework for determining the congruences on a number of monoids and categories of transformations, diagrams, matrices and braids, and on all their ideals. The key theoretical advances present ...