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

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 Bn and the Temperley-Lieb monoid T Ln.