Properties of congruences of twisted partition monoids and their lattices

TL;DR

This paper investigates the algebraic and order-theoretic properties of congruences on twisted partition monoids, revealing their generation, lattice structure, and counting formulas, with implications for algebraic theory.

Contribution

It characterizes the generation, lattice properties, and enumeration of congruences on twisted partition monoids, extending understanding of their algebraic and order-theoretic structure.

Findings

Each congruence is generated by at most .5n pairs.

The congruence lattice is not modular or distributive.

Number of congruences has a rational generating function.

Abstract

We build on the recent characterisation of congruences on the infinite twisted partition monoids $\mathcal{P}_{n}^\Phi$ and their finite $d$-twisted homomorphic images $\mathcal{P}_{n,d}^\Phi$, and investigate their algebraic and order-theoretic properties. We prove that each congruence of $\mathcal{P}_{n}^\Phi$ is (finitely) generated by at most $\lceil\frac{5n}2\rceil$ pairs, and we characterise the principal ones. We also prove that the congruence lattice $\textsf{Cong}(\mathcal{P}_{n}^\Phi)$ 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 $\textsf{Cong}(\mathcal{P}_{n,d}^\Phi)$ is modular but still not distributive for $d>0$, while $\textsf{Cong}(\mathcal{P}_{n,0}^\Phi)$ is distributive. We also calculate the number of congruences of $\mathcal{P}_{n,d}^\Phi$, showing that the array $\big(|\textsf{Cong}(\mathcal{P}_{n,d}^\Phi)|\big)_{n,d\geq 0}$ has a rational generating function, and that for a fixed $n$ or $d$, $|\textsf{Cong}(\mathcal{P}_{n,d}^\Phi)|$ is a polynomial in $d$ or $n\geq 4$, respectively.