Abstract
A major result of D.B. McAlister for inverse semigroups is generalised in the paper to classes of regular semigroups, including the class of all regular semigroups. It is shown that any regular semigroup is a homomorphic image of a regular semigroup whose least full self-conjugate subsemigroup is unitary; the homomorphism is injective on the subsemigroup. As an application, the group complexity of any finite E-solid regular semigroup is shown to be the same as, or one more than that of its least full self-conjugate subsemigroup (the subsemigroup is completely regular and is the type II subsemigroup). In an addition to the paper, by P.R. Jones, it is shown that any finite locally orthodox semigroup has group complexity 0 or 1.
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