The Wagner-Preston theorem is a generalization of Cayley's theorem to inverse semigroups. The same idea, when sets have additional local structure, is responsible for the appearance of pseudogroups of symmetries in geometry.

Statement

Every inverse semigroup has an injective homomorphism into the inverse semigroup of partial bijections of some set.

Literature

V. V. Wagner, Generalised groups. Proceedings of the USSR Academy of Sciences (in Russian) 84 (1952), 1119–1122.

V. V. Wagner, The theory of generalised heaps and generalised groups, Matematicheskii Sbornik. Novaya Seriya (in Russian). 32 (74) (1953): 545–632.

G. B. Preston, Representation of inverse semi-groups, J. London Math. Soc. 29 (1954) 411-419