Dimensions of Cantor and post varieties

A dimension of a finitely based variety V of algebras is the greatest length of a basis (that is, an independent generating set) for the SC-theory SC(V) with the strong Mal'tsev conditions satisfied in V. A dimension is said to be infinite if the lengths of bases in SC(V) are unbounded. We prove that the dimension of a Cantor variety C_m,n in the general form, i.e., with n>m≥1, is infinite. The algorithm of constructing a basis of any given length in SC(C_m,n) is presented. By contrast, any Post variety P_n generated by a primal algebra of order n>1 is shown to have a finite dimension not exceeding the number of maximal subalgebras in the iterative Post algebra over the set {0,1,…,n−1}. Specifically, the dimension of the variety of Boolean algebras is at most four. Translated fromAlgebra i Logika, Vol. 35, No. 3, pp. 359–369, May–June, 1996.