Classification of congruences of twisted partition monoids

The twisted partition monoid P^Φ_n is an infinite monoid obtained from the classical finite partition monoid 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 P^Φ_n. The succinct encoding of a congruence, which we call a C-pair, consists of a sequence of n+1 congruences on the additive monoid N of natural numbers and a certain (n+1)×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 P^Φ_n,d, which are obtained by factoring out from P^Φ_n 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.