Abstract. In this paper we prove that every $E$-unitary cover of an orthodox semigroup with a regular band of idempotents arises from a close embedding into an almost factorizable orthodox semigroup. Since every orthodox semigroup has an $E$-unitary cover, this shows that every orthodox semigroup with a regular band of idempotents is closely embeddable into an almost factorizable orthodox semigroup.

AMS Subject Classification (1991): 20M19, 20M10

Received December 8, 2006, and in revised form January 29, 2007. (Registered under 5955/2009.)