Abstract
If R is an ~-class of a semigroup S, then R inherits the structure of a partial groupoid from the binary operation on S, the product of two elements a,b of R being defined to be their product ab in S if abe R, and being undefined otherwise. The question arises how to tell when a given partial groupoid is isomorphic to an ~class of some semigroup. It is the purpose of this paper to answer this question.
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