  In 1995 Grillet defined the concept of a stratified semigroup and a
stratified semigroup with zero. The present authors extended that idea to
include semigroups with a more general base and proved, amongst other things,
that finite semigroups in which the H-classes contain idempotents, are
semilattices of stratified extensions of completely simple semigroups, and
every strict stratified extension of a Clifford semigroup is a semilattice of
stratified extensions of groups. We continue this work here by considering the
multiplicative semigroup of Dedekind domains and show in particular that
quotients of such rings have a multiplicative structure that is a (finite)
Boolean algebra of stratified extensions of groups.
