Classification of congruences of twisted partition monoids

James East, Nik Ruskuc

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)
6 Downloads (Pure)


The twisted partition monoid PΦn 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 PΦn. 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 PΦn,d, which are obtained by factoring out from PΦn 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.
Original languageEnglish
Number of pages65
JournalAdvances in Mathematics
VolumeIn Press
Early online date18 Nov 2021
Publication statusE-pub ahead of print - 18 Nov 2021


  • Partition monoid
  • Twisted partition monoid
  • Congruence
  • Congruence lattice


Dive into the research topics of 'Classification of congruences of twisted partition monoids'. Together they form a unique fingerprint.

Cite this