Homomorphisms of a Semigroup Onto a Group
Robert R. Stoll · American Journal of Mathematics · 1951
asked in connection with images of the form, a group with a zero element adjoined. These images exist if and only if S contain a prime ideal, according to Theorem 1. In the present paper the notions of a maximal group image and maximal group with zero image are formulated for a semigroup S and the construction of such images is discussed in terms of subsystems (generalizations of a normal subgroup in the group case) of S, using a method devised by Dubreil [2]. A set of independent defining conditions for these normal subsystems is obtained. Finally, several well known semigroups which have unique maximal images of the types under considerations are discussed. The most interesting examples are two classes of semigroups (regular sets of partial transformations of a set, and completely simple semigroups without zero) studied by Rees in [7] and [5] respectively, and one (semigroups having zeroid elements) studied by Clifford and Miller in [1]. For the first class, the group image discussed by Rees is shown to be maximal. For the second class, the maximal group image can be described in terms of a homomorphic image of the basis group for the semigroup. For the third class; the group of zeroid elements is found to be the maximal