Classification of congruences of twisted partition monoids

The twisted partition monoid PnΦ is an infinite monoid obtained from the classical finite partition monoid Pn 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 PnΦ. 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) x 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 Pn,dΦ, which are obtained by factoring out from PnΦ 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.