Generalised Groups

Abstract

In this short communication to the Academy of Sciences, Wagner considered semigroups S in which for every element s there is a corresponding element s ^′ such that ss ^′ s = s and s ^′ ss ^′ = s ^′. If, in addition, the idempotents commute in such a semigroup, then the element s ^′ is unique, is termed the generalised inverse of s and is denoted s ⁻¹. After proving basic results about generalised inverses, Wagner defined a generalised group (or inverse semigroup in modern terminology) to be a semigroup in which every element has a unique generalised inverse. He further defined a natural partial order in such a semigroup and derived certain of its basic properties. Turning to semigroups of partial transformations, Wagner noted that the semigroup of all one-to-one partial transformations of a set forms a generalised group, and, moreover, that any generalised group may be represented as one of these, a result now termed the Wagner–Preston Representation Theorem.