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

Matthew Brookes, James East, Craig Miller, James D. Mitchell and Nik Ruškuc

Vol. 333 (2024), No. 1, 17–57

DOI: 10.2140/pjm.2024.333.17

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 ${\mathcal{𝒮}}_n$ is known, as is an accurate asymptotic formula for the height of the subsemigroup lattice of the full transformation monoid ${\mathcal{𝒯}}_n$. Motivated by the related question of determining the heights of the lattices of left and right congruences of ${\mathcal{𝒯}}_n$, 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 ${\mathcal{𝒯}}_n$, the partial transformation monoid $\mathcal{𝒫}{\mathcal{𝒯}}_n$, the symmetric inverse monoid ${\mathcal{ℐ}}_n$, the monoid of order-preserving transformations ${\mathcal{𝒪}}_n$, the full matrix monoid $\mathcal{ℳ}(n,q)$, the partition monoid ${\mathcal{𝒫}}_n$, the Brauer monoid ${\mathcal{ℬ}}_n$ and the Temperley–Lieb monoid $\mathcal{𝒯}{L}_n$.

Keywords

semigroup, semigroup action, unary algebra, (left/right) congruence, congruence lattice, height, modular element, transformation and diagram monoids, Schützenberger group

Mathematical Subject Classification

Primary: 20M10

Secondary: 06B05, 08A30, 08A60, 20M20, 20M30

Milestones

Received: 5 December 2023

Revised: 18 November 2024

Accepted: 30 November 2024

Published: 19 December 2024