{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  ∈  Q and if a  ≠  b , a $\circ$ b  =  c and b $\circ$ a  =  d if and only if the m ‐cycle (…, d , x , a , b , y , c , …) ∈  C ). The object of this paper is to show that the class of {2,3}‐perfect m ‐cycle systems cannot be equationally defined for all m  ≥ 10, m  ≠ 11. This combined with previous results shows that {2, 3}‐perfect m ‐cycle systems are equationally defined for m  = 5, 7, 8, 9, and 11 only. © 2004 Wiley Periodicals, Inc.