A connection between varieties of quasigroups and graph decompositions Original Research Article

In this paper we define an infinite family of varieties of algebraic quasigroups, where for m⩾3 the variety Vm satisfies a particular set of defining identities Im. It is shown that the finite members of Vm are precisely the quasigroups that can be obtained via a standard construction from certain decompositions of a complete graph into closed trails of lengths that each divide m. The result is presented from a graph theoretical perspective.