A semigroup S is called .E-inversive if for every a € S there is an x € S such that (ax) 1 = ax. A construction of all £-inversive subdirect products of two £-inversive semigroups is given using the concept of subhomomorphism introduced by McAlister and Reilly for inverse semigroups. As an application, E-unitary covers for an is-inversive semigroup are found, in particular for those whose maximum group homomorphic image is a given group. For this purpose, the explicit form of the least group

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... the least group congruence on an arbitrary .E-inversive semigroup is given. The special case of full subdirect products of a semilattice and a group (that is, containing all idempotents of the direct product) is investigated and, following an idea of Petrich, a construction of all these semigroups is provided. Finally, all periodic semigroups which are subdirect products of a semilattice or a band with a group are characterized.