Properties of congruences of twisted partition monoids and their lattices

James East, Nik Ruškuc

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)

Abstract

We build on the recent characterisation of congruences on the infinite twisted partition monoids (Formula presented.) and their finite (Formula presented.) -twisted homomorphic images (Formula presented.), and investigate their algebraic and order-theoretic properties. We prove that each congruence of (Formula presented.) is (finitely) generated by at most (Formula presented.) pairs, and we characterise the principal ones. We also prove that the congruence lattice (Formula presented.) is not modular (or distributive); it has no infinite ascending chains, but it does have infinite descending chains and infinite anti-chains. By way of contrast, the lattice (Formula presented.) is modular but still not distributive for (Formula presented.), while (Formula presented.) is distributive. We also calculate the number of congruences of (Formula presented.), showing that the array (Formula presented.) has a rational generating function, and that for a fixed (Formula presented.) or (Formula presented.), (Formula presented.) is a polynomial in (Formula presented.) or (Formula presented.), respectively.
Original languageEnglish
Pages (from-to)311-357
Number of pages47
JournalJournal of the London Mathematical Society
Volume106
Issue number1
DOIs
Publication statusPublished - 2022

Open Access - Access Right Statement

© 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 (https://creativecommons.org/licenses/by/4.0/), which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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