Groupoids determined by involutive automorphisms on semilattices of groups

R. A. R. Monzo

Abstract: We explore properties of groupoids $(S,\cdot)$ of the form $x\cdot y=(\alpha x)* y$, for all $\{x,y\}\subseteq S$, where $(S,*)$ is a semigroup and a semilattice of groups. We require $\alpha$ to be an idempotent-fixed automorphism on $(S, *)$ whose square is the identity map. Several characterizations of such groupoids are proved, using analogies of semigroup-theoretic properties and constructions.