Heights of one- and two-sided congruence lattices of semigroups

Matthew Brookes, James East, Craig Miller, James David Mitchell, Nik Ruskuc*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

The height of a poset P is the supremum of the cardinalities of chains in P. The exact formula for the height of the subgroup lattice of the symmetric group Sn is known, as is an accurate asymptotic formula for the height of the subsemigroup lattice of the full transformation monoid Tn. Motivated by the related question of determining the heights of the lattices of left and right congruences of Tn, and deploying the framework of unary algebras and semigroup actions, we develop a general method for computing the heights of lattices of both one- and two-sided congruences for semigroups. We apply this theory to obtain exact height formulae for several monoids of transformations, matrices and partitions, including the full transformation monoid Tn, the partial transformation monoid PTn, the symmetric inverse monoid In, the monoid of order-preserving transformations On, the full matrix monoid M(n, q), the partition monoid Pn, the Brauer monoid Bnand the Temperley–Lieb monoid T Ln.
Original languageEnglish
Pages (from-to)17-58
Number of pages41
JournalPacific Journal of Mathematics
Volume333
Issue number1
Early online date19 Dec 2024
DOIs
Publication statusPublished - 2024

Keywords

  • Semigroup
  • Semigroup action
  • Unary algebra
  • (Left/right) congruence
  • Congruence lattice
  • Height
  • Modular element
  • Transformation and diagram monoids
  • Schützenberger group

Fingerprint

Dive into the research topics of 'Heights of one- and two-sided congruence lattices of semigroups'. Together they form a unique fingerprint.

Cite this