{2, 3}-Perfect m-Cycle Systems Are Equationally Defined for m=5, 7, 8, 9,and 11 Only

An m-cycle system (S;C) of order n is said to be {2,3}-perfect provided each pair of vertices is connected by a path of length 2 in an m-cycle of C and a path of length 3 in an m-cycle of C. The class of {2,3}-perfect m-cycle systems is said to be equationally defined provided,there exists a variety of quasigroups V with the property that a finite quasigroup (Q; \circ; \;/) belongs to V if and only if its multiplicative (Q; \circ) part can be constructed from a {2,3}-perfect m-cycle system using the 2-construction (a \circ a = a for all a \in Q and if a \neq b, a \circ b = c andb \circ a = d if and only if the m-cycle (. . . ; d; x; a; b; y; c; . . .) \in C). The object of this paper is toshow that the class of {2,3}-perfect m-cycle systems cannot be equationally defined for allm \leq 10;m \neq 11. This combined with previous results shows that {2,3}-perfect m-cycle systemsare equationally defined for m = 5, 7, 8, 9, and 11 only.