VARIETIES DEFINED BY PERMUTATIONS

Let Sn be a symmetric group on a set {1,2,... ,n}. For an arbitrary permutation (pi) of Sn, we consider a variety nG(pi) of n-groupoids (A,f) satisfying the identity f(x1,x2,...,xn) = f(x(pi)(1),x(pi)(2),. . . ,x(pi)(n)). It is proved that if lengths of all independent cycles of (pi) are positive of one number m >,= 2 then nG(pi) has a finite dimension equal to the number of prime divisors of m. The dimension of a variety; in this event, is the least upper bound of lengths of independent bases for the collection of all strong Malʹtsev conditions satisfied in that variety.