Classification of congruences of twisted partition monoids

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