  We introduce the depth parameters of a finite semigroup, which measure how
hard it is to produce an element in the minimum ideal when we consider
generating sets satisfying some minimality conditions. We estimate such
parameters for some families of finite semigroups, and we obtain an upper bound
for wreath products and direct products of two finite (transformation) monoids.
Keywords: semigroup, generating set, minimum ideal, A-depth of a semigroup
