# Moscow Mathematical Journal

Volume 22, Issue 2, April–June 2022 pp. 295–372.

Congruences on Infinite Partition and Partial Brauer Monoids

**Authors**:
James East (1) and Nik Ruškuc (2)

**Author institution:**(1) Centre for Research in Mathematics, School of Computing, Engineering and Mathematics, Western Sydney University, Locked Bag 1797, Penrith NSW 2751, Australia

(2) Mathematical Institute, School of Mathematics and Statistics, University of St Andrews, St Andrews, Fife KY16 9SS, UK

**Summary: **

We give a complete description of the congruences on the partition monoid $\mathcal{P}_X$ and the partial Brauer monoid $\mathcal{PB}_X$, 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, Ruškuc and Torpey, which deals with the finite case. As a consequence of our classification result, we show that the congruence lattices of $\mathcal{P}_X$ and $\mathcal{PB}_X$ 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.

2020 Math. Subj. Class. 20M20, 08A30, 06A06, 03E04.

**Keywords:**Diagram monoids, partition monoids, partial Brauer monoids, congruences, well quasi-orderedness.

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