Representation of inverse semigroups by local automorphisms and multi- automorphisms of groups and rings (Q1057991)

A local automorphism of an algebra \({\mathfrak A}\) is any isomorphism between two (possibly empty) subalgebras of \({\mathfrak A}\). A multivalued mapping of \({\mathfrak A}\) into itself is called a multi-automorphism of \({\mathfrak A}\) if this mapping preserves the operations of \({\mathfrak A}\), and each element of \({\mathfrak A}\) has a nonempty set of images and a nonempty set of inverse images under this multivalued mapping. For many algebras (groups, rings, Boolean algebras, etc.) the set of all multi-automorphisms forms an inverse semigroup. The note contains an almost instant proof of the fact that a (finite) inverse semigroup is isomorphic to an inverse semigroup of local automorphisms of a (finite) group, commutative ring, a Boolean algebra, a vector space, etc. This result (without finiteness assumptions and with a much more complicated proof) was published by \textit{B. Fichtner-Schultz} [Math. Nachr. 44, 313-339 (1970; Zbl 0176.294), and 48, 275-278 (1971; Zbl 0206.025)]. It has been also proved in the Doctor of Sciences dissertation of the author of 1965. An analogous theorem (with a less trivial proof) holds if ''local automorphisms'' are replaced by ''multi- automorphisms in the above result.