On equationally definable m-cycle systems

A class $C$ of $m$-cycle systems is said to be equationally defined provided there exists a variety of algebraic quasigroups $V$ such that a finite quasigroup belongs to $V$ if and only if the multiplicative part of the quasigroup can be constructed from a $m$-cycle system belonging to $C$ using a given construction (too complicate to go into here). In [3] it is shown that the class of $2$-perfect $m$-cycle systems can be equationally defined for $m=3,5,$ and$7$ only. In [5] it is shown that the class of $2,3$-perfect $7$-cycle systems can be equationally defined. In this paper we extend this result by showing that the class of $2,3$-perfect $m$-cycle systems can be equationally defined for $m=8,9,$ and $11$. In each case we give a defining set of identities for the variety.