# Quasigroups constructed from cycle systems

An m-cycle system of order n is a pair (Q,C), where C is a collection of edge-disjoint m-cycles which partitions the edge set of the complete undirected graph K_n with vertex set Q. If the m-cycle system (Q,C) has the additional property that every pair of vertices {a,b} is joined by a path of length 2 (and therefore exactly one) in an m-cycle of C, then (Q,C) is said to be 2-perfect.

Now given an m-cycle system (Q,C), we can define a binary operation * on Q by a * a = a and if {a,b} is a pair of vertices, by a * b = c, b * a = d, if and only if the cycle (... , d, a, b, c, ...) belongs to C. This is called the Standard Construction and it is well-known that the groupoid (Q,*) is a quasigroup [which can be considered to be the ``multiplicative'' part of the universal algebra quasigroup (Q, *, \, /)] if and only if (Q,C) is 2-perfect.

The class of 2-perfect m-cycle systems is said to be equationally defined if and only if there exists a variety of quasigroups V such that the finite members of V are precisely all universal algebra quasigroups whose multiplicative parts can be constructed from 2-perfect m-cycle systems using the Standard Construction. For example, 3-cycle systems (or Steiner triple systems, which are always 2-perfect) are equationally defined since the quasigroup variety V defined by the identities I = { x^2=x, xy=yx, (yx)x=y } has the property that a finite quasigroup (Q, *, \, /) belongs to V if and only if (Q,*) can be constructed from a Steiner triple system using the Standard Construction.

This talk gives a survey of results showing that the class of 2-perfect m-cycle systems can be equationally defined for m = 3, 5 and 7 only.