Embeddable partial groups

Abstract

We record a folklore theorem that says a partial group embeds in a group if and only if each word has at most one possible multiplication, regardless of choice of parenthesization. We further investigate the partial groups which are exemplars of non-embeddability. Finally we show that a partial groupoid embeds in a groupoid if and only if its reduction embeds in a group.

13 pages. v2: add alternate characterization of degree of universal counterexamples to section 3 (and other minor adjustments)