Classification of congruences of twisted partition monoids

The twisted partition monoid $\mathcal{P}_n^\Phi$ is an infinite monoid obtained from the classical finite partition monoid $\mathcal{P}_n$ by taking into account the number of floating components when multiplying partitions. The main result of this paper is a complete description of the congruences on $\mathcal{P}_n^\Phi$. The succinct encoding of a congruence, which we call a C-pair, consists of a sequence of $n+1$ congruences on the additive monoid $\mathbb{N}$ of natural numbers and a certain $(n+1)\times\mathbb{N}$ matrix. We also give a description of the inclusion ordering of congruences in terms of a lexicographic-like ordering on C-pairs. This is then used to classify congruences on the finite $d$-twisted partition monoids $\mathcal{P}_{n,d}^\Phi$, which are obtained by factoring out from $\mathcal{P}_n^\Phi$ the ideal of all partitions with more than $d$ floating components. Further applications of our results, elucidating the structure and properties of the congruence lattices of the ($d$-)twisted partition monoids, will be the subject of a future article.