The spectra of a variety of quasigroups and related combinatorial designs

Abstract

A quasigroup is an ordered pair ( Q,·), where Q is a set and (·) is a binary operation on Q such that the equations ax = b and ya = b are uniquely solvable for every pair of elements a,b in Q. It is well-known that the multiplication table of a quasigroup defines a Latin square, and to each quasigroup are associated six (not necessarily distinct) conjugate quasigroups. The spectrum of the two-variable quasigroup identity u( x, y) = v( x, y) is the set of all integers n such that there exists a quasigroup of order n satisfying the identity u( x, y) = v( x, y). Trevor Evans has provided a collection of two-variable quasigroup identities, which imply that two conjugates are orthogonal and which are conjugate-equivalent to “short conjugate-orthogonal identities”. These identities include the familiar Stein identity, x( xy) = yx, which has been given a considerable amount of attention. Apart from being associated with conjugate orthogonal Latin squares, some of the identities have been used in the description of other types of combinatorial designs, such as BIBDs, Mendelsohn designs, certain classes of graphs, and orthogonal arrays with interesting conjugacy properties. We shall briefly survey the known results and in some cases we present new results concerning the spectra of the short conjugate-orthogonal identities, which have not been previously investigated. The emphasis will be on the constructions and uses of pairwise balanced designs (PBDs) and related combinatorial structures.