  We investigate the maximum length of a chain of subsemigroups in various
classes of semigroups, such as the full transformation semigroups, the general
linear semigroups, and the semigroups of order-preserving transformations of
finite chains. In some cases, we give lower bounds for the total number of
subsemigroups of these semigroups. We give general results for finite
completely regular and finite inverse semigroups. Wherever possible, we state
our results in the greatest generality; in particular, we include infinite
semigroups where the result is true for these. The length of a subgroup chain
in a group is bounded by the logarithm of the group order. This fails for
semigroups, but it is perhaps surprising that there is a lower bound for the
length of a subsemigroup chain in the full transformation semigroup which is a
constant multiple of the semigroup order.
