A Class of d-Simple Semigroups
A. H. Clifford · American Journal of Mathematics · 1953
Conversely, if P is any semigroup having properties Bi, 2, 3, tllere exists a semigroup S satisfying Al, 2, 3, the right unit subsemigroup of which is isomorphic with P. A detailed statement of this result is given in ? 1, and the rest of the paper is devoted to the proof thereof. The problem of describing all semigroups S satisfying Al, 2, 3 is of course merely replaced by the perhaps equally difficult one of describilng all semigroups P satisfying BI, 2, 3. The extent of the class of all such P can be seen from the fact that the positive part of a lattice-ordered group ( [1], Chapter 14) satisfies not only B1, 2, 3 but also the left-right duals thereof. The present work parallels rather than extends that of Suschkewitsch and IRees. For while a completely simple semigroup is regular and d-simple, it reduces to a group if either of the conditions A2 or A3 is imposed upon it.