Congruence lattices of powers of an algebra
Let \(\mathcal P\) be a property which can be attributed to 0-1 sublattices of the equivalence lattice on an arbitrary set. It was already proved by S. Burris and the author that for every integer \(k\geq 2\) there is an \(n\geq 1\) such that for every algebra \(A\) of cardinality \(k\), if \(\text{Con}(A^ m)\) satisfies \(\mathcal P\) for all \(m\leq n\) then \(\text{Con}(A^ m)\) satisfies \(P\) for all \(m\). Let \(n_{\mathcal P}(k)\) denote the least such \(n\). The paper evaluates the function \(n_{\mathcal P}\) for \(p=\) permutability, distributivity, arithmeticity, weak distributivity, modularity, and the property of being skew-free (= the Fraser-Horn property). Main results: (i) \(n_{\text{Perm}}(k)\leq k^ 3\); (ii) \(n_{\text{Dist}}(k)\leq k^{k+1}\); (iii) \(n_{\text{Skew-free}}(k)\leq k^3 + k^2 - k\); (iv) \(n_{\text{Mod}}(k)\leq k^{4m(k)}\), where \(m(k)=k^{(k^ 4-k^3 + k^2)} - 1\).
scientific article; zbMATH DE number 4123795