  In this paper we present a new embedding of a semigroup into a semiband
(idempotent-generated semigroup) of depth 4 (every element is the product of 4
idempotents) using a semidirect product construction. Our embedding does not
assume that S is a monoid (although it assumes a weaker condition), and works
also for (non-monoid) regular semigroups. In fact, this semidirect product is
particularly useful for regular semigroups since we can defined another
embedding for these semigroups into a smaller semiband of depth 2. We shall
then compare our construction with other known embeddings, and we shall see
that some properties of S are preserved by our embedding.
