Heights of one- and two-sided congruence lattices of semigroups
Matthew Brookes, James East, Craig J. Miller, James D. Mitchell, Nik Ruškuc · Pacific Journal of Mathematics · 2024
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 S n is known, as is an accurate asymptotic formula for the height of the subsemigroup lattice of the full transformation monoid T n .Motivated by the related question of determining the heights of the lattices of left and right congruences of 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 T n , the partial transformation monoid PT n , the symmetric inverse monoid I n , the monoid of order-preserving transformations O n , the full matrix monoid M(n, q), the partition monoid P n , the Brauer monoid B n and the Temperley-Lieb monoid T L n .The authors are supported by the Engineering and Physical Sciences Research Council [EP/S020616/1, EP/V002953/1 and EP/V003224/1] and the Australian Research Council [FT190100632].