Congruences on infinite partition and partial Brauer monoids

Abstract

We give a complete description of the congruences on the partition monoid $P_X$ and the partial Brauer monoid $P{B}_X$, 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 $P{B}_X$ 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.

arXiv commentTo appear in Moscow Math J. V2: 69 pages, 11 figures, 1 table, expanded introduction, more references, incorporates referee's suggestions. V1: 66 pages, 11 figures, 1 table

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