Congruences on infinite partition and partial Brauer monoids

TL;DR

This paper classifies all congruences on the infinite partition and partial Brauer monoids, revealing their lattice structures and properties, and extends finite case results to infinite sets.

Contribution

It provides a complete description of congruences on infinite partition and partial Brauer monoids, including their lattice structures and generation properties.

Findings

Congruence lattices are isomorphic for both monoids.

Lattices are distributive and well quasi-ordered.

Number of pairs needed to generate congruences can be infinite, depending on cofinality.

Abstract

We give a complete description of the congruences on the partition monoid $P_X$ and the partial Brauer monoid $PB_X$, where $X$ is an arbitrary infinite set, and also of the lattices formed by all such congruences. Our results complement those from a recent article of East, Mitchell, Ruskuc and Torpey, which deals with the finite case. As a consequence of our classification result, we show that the congruence lattices of $P_X$ and $PB_X$ are isomorphic to each other, and are distributive and well quasi-ordered. We also calculate the smallest number of pairs of partitions required to generate any congruence; when this number is infinite, it depends on the cofinality of certain limit cardinals.