On the Structure of Partial Automorphism Semigroups

Let S be a set and ∑ a mathematical structure on S. If a subset of S is distinguished by a certain property τ = τ(∑) we call it a τ-subset of S. Let Θ be some reduct of ∑. Any Θ-isomorphism between two τ-subsets of S is called a partial τΘ-automorphism of S (in some cases, e.g., for geometric structures, it is natural to use the word “local” instead of “partial”). For example, let (S, ∑) be a topologial space, let the τ-subsets of S be precisely the ∑-open sets, and let θ be just E; then partial τΘ-automorphisms are exactly the homeomorphisms between open subsets of S. As another example, consider a universal algebra (S, ∑). Let τ- subsets of S be the finitely generated ∑-subalgebras of S and let Θ be a certain reduct of ∑. Then partial τΘ-automorphisms of S are the Θ-isomorphisms between finitely generated subalgebras of S. (The last example could be specialized, say, to a ring (S, +,·,0) with Θ = {·}; then partial τΘ-automorphisms of S are multiplicative isomorphisms between finitely generated subrings of S.) A number of other examples will be considered later in the paper.