Congruence lattices of powers of an algebra. (English) Zbl 0686.08008

Willard, Ross

Algebra Univers. 26, No. 3, 332-340 (1989).

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 \(m(k)=k^{(k^ 4-k^3 + k^2)} - 1\).

Reviewer: J.Duda

MSC:

08B10	Congruence modularity, congruence distributivity
08A30	Subalgebras, congruence relations

Keywords:

power of algebra; congruence lattice; permutability; distributivity; arithmeticity; weak distributivity; modularity; skew-free; Fraser-Horn property

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