Congruence lattices of powers of an algebra

Let ~ be a property which can be attributed to 0, 1-sublattices of Eq (X), X arbitrary. (Eq (X) is the lattice of all equivalence relations on X.) For example, could be modularity or permutability. It was proved in [3] that for every integer k 2 there is an n -> 1 such that for every algebra A of cardinality k, if Con (A m) satisfies ~ for all m -< n then Con (A m) satisfies ~ for all m. Let ne(k) denote the least such n. The open problem is: given N, to evaluate the function n~. In this paper we establish an upper bound to n~ for each of the following properties ~: permutability, distributivity, modularity and the property of being skew-flee. The upper bounds established here are much too large; for example, the bound for modularity is