Extending partial isomorphisms and McAlister's covering theorem (Q1313472)

Inverse semigroups are the abstract counterparts of the semigroups of partial one-to-one structure preserving functions of many mathematical structures. Given such a semigroup \(\Gamma(\Sigma)\) of partial functions of a structure \(\Sigma\) it is a natural question whether \(\Sigma\) is a reduct of a structure \(\Sigma'\) of the same type with the property that every element of \(\Gamma(\Sigma)\) is the restriction of an automorphism of \(\Sigma'\). Thus \(\Sigma'\) is a (generalised) HNN-extension of \(\Sigma\). In this paper, it is shown how the abstract relationship between \(\Gamma(\Sigma)\) and \(\Gamma(\Sigma')\) can be formalised within the theory of ordered groupoids, and that this leads naturally to the definition of McAlister triples, and the existence of \(E\)-unitary covers and factorisable embeddings.