  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 $\mathcal{T}_n$, 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 $\mathcal{PT}_n$, 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 $\mathcal{TL}_n$.
