On a pull-back diagram for orthodox semigroups

In the present paper we deal with two problems concerning orthodox semigroups. M. Yamada raised the questions in [6] whether there exists an orthodox semigroup T with band of idempotents E and greatest inverse semigroup homomorphic image S for every band E and inverse semigroup S which have the property that is isomorphic to the semilattice of idempotents of S, and if T exists then whether it is always unique up to isomorphism. T. E. Hall [1] has published counter-examples in connection with both questions and, moreover, he has given a necessary and sufficient condition for existence. Now we prove a more effective necessary and sufficient condition for existence and deal with uniqueness, too. On the other hand, D. B. McAlister's theorem in [4] saying that every inverse semigroup is an idempotent separating homomorphic image of a proper inverse semigroup is generalized for orthodox semigroups. The proofs of these results are based on a theorem concerning a special type of pullback diagrams. In verifying this theorem we make use of the results in [5] which we draw up in Section 1. The main theorems are stated in Section 2. For the undefined notions and notations the reader is referred to [2].