Upper Bounds On The Sizes Of Finitely Generated Algebras

We present an upper bound for the cardinality of any n-generated algebra in a locally finite variety V of algebras. This upper bound depends only on some fundamental numerical invariants of the n-generated subdirectly irreducible algebras in V. A theorem characterizing those varieties that contain algebras whose cardinalities achieve the upper bound is proved. Several explicit methods for computing the exact values of these invariants are described. The final section contains detailed concrete examples illustrating applications of the characterization theorem and of the various methods for computing the upper bound.