Injectives in varieties of completely regular semigroups

Abstract

The injectives in some varieties of completely regular semigroups are known. In particular the injectives are known in the following varieties: any variety of bands (Gerhard [2]); any variety of groups (Garcia-Larri6n [l] and Kovacs-Newman [6]); any variety of completely simple semigroups (Trotter [12]); and the variety of all semilattices of abelian groups (Schein [ll]). In this paper the semigroups of the title will be specified. The paper begins with definitions and statements of the relevant results from [l, 2,6, 121. In the next section it is shown that any injective in a variety of completely regular semigroups is an orthodox normal band of groups with retractions for its structure maps. In Section 3 the investigation is restricted to semilattices of groups that are injectives in a variety Yof completely regular semigroups. Necessary and sufficient conditions for a semilattice of groups to be a “Y-injective are obtained. In Section 4 the semilattice of groups results are generalised so that a description of the remaining injectives in “Y can be given.