A variety \(V\) of quasigroups can be equationally defined if a collection of quasigroup identities exists such that \(V\) is precisely the set of all quasigroups satisfying these identities. An \(m\)-cycle system \((S,C)\) of order \(n\) is 2-perfect if the collection of distance 2 graphs of \(C\) also covers the edge set of \(K_n\). One can construct a quasigroup of order \(n\) from an \(m\)-cycle system \((S,C)\) of order \(n\) by defining a binary operation \(\circ\) on \(S\) as follows: (1) \(x \circ x = x\), for all \(x \in S\); and (2) if \(x \neq y\), \(x \circ y = z\) and \(y \circ x = w\) if and only if \((\dots, w,x,y,z, \dots) \in C\). This is called the standard construction. The class of 2-perfect \(m\)-cycle systems is equationally defined if and only if a variety of quasigroups \(V\) exists with the property that the finite quasigroups in \(V\) are precisely the quasigroups whose multiplicative parts can be constructed from 2-perfect \(m\)-cycle systems using the standard construction.
In this paper, the author draws correlations between universal algebra and graph theory in a manner which is clear, interesting and, at times, quite entertaining to read. More specifically, he begins by giving the classical correspondence between certain groupoids and quasigroups and cycle systems. The remainder of the paper considers the following question: “For which \(m \geq 3\) is the class of 2-perfect \(m\)-cycle systems equationally defined?” The author surveys the recent work which has shown that the class of 2-perfect \(m\)-cycle systems can be equationally defined for \(m=3,5\) and 7 only.