Maximal n-generated subdirect products

For n a positive integer and K a finite set of finite algebras, let L(n,K) denote the largest n-generated subdirect product whose subdirect factors are algebras in K. When K is the set of all ngenerated subdirectly irreducible algebras in a locally finite variety V, then L(n,K) is the free algebra FV(n) on n free generators for V. For a finite algebra A the algebra L(n, {A}) is the largest n-generated subdirect power of A. For every n and finite A we provide an upper bound on the cardinality of L(n, {A}). This upper bound depends only on n and these basic parameters: the cardinality of the automorphism group of A, the cardinalities of the subalgebras of A, and the cardinalities of the equivalence classes of certain equivalence relations arising from congruence relations of A. Using this upper bound on n-generated subdirect powers of A, as A ranges over the n-generated subdirectly irreducible algebras in V, we obtain an upper bound on |FV(n)|. And if all the n-generated subdirectly irreducible algebras in V have congruence lattices that are chains, then we characterize in several ways those V for which this upper bound is obtained.