A connection between varieties of quasigroups and graph decompositions

In this paper we define an infinite family of varieties of algebraic quasigroups, where for m greater than or equal to 3 the variety V-m satisfies a particular set of defining identities I-m. It is shown that the finite members of V-m 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. (C) 2003 Elsevier B.V. All rights reserved.