  In 1995 Grillet introduced the concept of a stratified semigroup as a kind of
generalisation of finite nilsemigroups. We extend these ideas here by allowing
a more general Base and describe them in terms of extensions of semigroups by
stratified semigroups. We consider semillatices of certain types of group-bound
semigroups and also semillatices of Clifford semigroups and show how to
describe them as semilattices of these stratified extensions and provide a
number of interesting examples.
