On orthodox semigroups determined by their bundles of correspondences (Q1207869)

It is known that the set \({\mathcal C}(S)\) of all correspondences of any given algebra \(S\) (i.e. stable binary relations on \(S\)) forms a partially ordered semigroup with involution with respect to the partial order ``\(\subseteq\)'', the composition ``\(\circ\)'' and the unary operation ``\(^{-1}\)''. The system \(({\mathcal C}(S),\circ,^{-1},\subseteq)\) is called the bundle of correspondences of \(S\). If \(S\) and \(S'\) have isomorphic bundles of correspondences, then they are called \(\mathcal C\)- isomorphic and any isomorphism \({\mathcal C}(S) \to {\mathcal C}(S')\) is called a \({\mathcal C}\)-isomorphism. The main result of the paper is the following Theorem: Let \(S\) be an orthodox semigroup whose maximum inverse semigroup morphic image is fundamental, and let \(S'\) be an arbitrary semigroup. If \(S\) and \(S'\) are \(\mathcal C\)-isomorphic, then any \(\mathcal C\)-isomorphism of \(S\) onto \(S'\) is induced by a unique bijection, and this bijection is an isomorphism or an antiisomorphism of \(S\) onto \(S'\).