title: Rectangular groupoids and related structures.
abstract: The quasivariety of groupoids [Formula: see text] satisfying the implication [Formula: see text] generalises rectangular semigroups and central groupoids. We call themrectangular groupoidsand find three combinatorial structures based upon arrays, matrices and graphs that are closely related. These generalise several groupoids of independent interest. The quasivariety generates the variety of all groupoids; they satisfy no nontrivial equations. We see some strong connections with isotopy, this being one of the classes of algebras (along with quasigroups) closed under isotopy. We investigate some constructions and show that a regular automorphism exists iff the groupoid is derived from a group via a Cayley graph construction.
authors.affiliation: Institute for Algebra, Johannes Kepler Universität Linz, Altenbergerstrasse 69, A-4040 Linz, Austria.
authors.forename: Tim
authors.initials: T
authors.lastname: Boykett
keywords: Dualities; General algebra; Groupoids; Isotopy; Matrix identities; Path property; Quasivarieties; Subvarieties; Transversals