Partial homomorphic images of Brandt groupoids

Abstract

The main purpose of the present note is to show (Theorem 2) that any regular D-class of any semigroup is a partial homomorphic image of a Brandt groupoid. It follows from this that a semigroup with zero is a partial homomorphic image of a Brandt semigroup if and only if it is regular and 0-bisimple. In the first section, an alternative formulation is given of the determination by H.-J. Hoehnke [1 ] of all partial homomorphisms of a Brandt groupoid into an arbitrary semigroup. This is first done (Theorem 1) for any completely 0-simple semigroup. The result is a straightforward generalization of Theorem 3.14 of [2], in which all partial homomorphisms of one completely 0-simple semigroup into another are determined. The present terminology is that of [2]; Hoehnke omits the adjective partial. Basic definitions given in [2] will not be repeated here; likewise, references to the fundamental work of Brandt, Rees, Green, and Munn can be found in [2].