Semigroups Admitting Relative Inverses

It is seen almost immediately (Theorem 1) that S admits relative inverses if and only if it is the class sum of mutually disjoint groups S., one to each idempotent element e of S. This fact tells us little about the structure of S since the product of two of these groups is not necessarily contained in a third, but may be scattered throughout several. (This occurs in the example given at the end of the paper, though simpler examples can be given.) It is shown in Theorem 2, however, that S is the class sum of mutually disjoint semigroups S,. of known structure such that no such scattering takes place. Each Sa is what Rees' calls a completely simple semigroup without zero. The structure of such a semigroup was, in the finite case, first given by Suschkewitsch,2 who calls it a Kerngruppe. Moreover it is possible to arrange these Sa in a semi-lattice3 P such that the product S,.So is contained in the greatest lower bound (in P)S,., of Sa and So . The structure of S is thus determined in the large, so to speak. In the special case in which any two idempotent elements of S commute with each other, the simple semigroups S,, reduce to groups, coinciding with the groups S,, and the semi-lattice P is isomorphic with the semigroup of all idempotent elements of S. In this case the structure of S is completely determined