Congruences on infinite partition and partial Brauer monoids

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.