Congruence lattices of finite algebras (Q1057300)

Using the theory of tame minimal sets developed by Ralph McKenzie, it is proved that all nonsimple finite algebras with tame minimal sets are abelian. Specifically, let \({\mathcal L}\) be the class of all bounded lattices L such that: i) The only congruence of L that identifies 0 or 1 with any other element is the universal congruence. ii) The only strictly increasing meet-endomorphism of L is the constant function with value 1. Then we have: Theorem: Let A be any nonsimple finite algebra with congruence lattice in \({\mathcal L}\). Then A satisfies the abelian term condition, that is, for all m and n, for all \((m+n)\)-ary polynomial functions f of A, all \(\bar a,\)\=b\(\in A^ m\) and all \(\bar c,\bar d\in A^ n\), \(f(\bar a,\bar c)=f(\bar a,\bar d)\) iff \(f(\bar b,\bar c)=f(\bar b,\bar d)\).