Congruence systems in dual discriminator varieties

Abstract

A congruence system on an algebra A is a tuple θ1,...,θk, a1,...,ak where θ1,...,θk ∈ Con A, a1,...,ak ∈ A and ai, aj ∈ θi ∨θj for all i, j ∈ {1,...,k}. A solution to such a congruence system is an element a ∈ A satisfying a, ai ∈ θi for all i ∈ {1,...,k}. A tuple of congruences θ1,...,θk is said to be a Chinese Remainder tuple (CR tuple for short) of A provided that every system θ1,...,θk, a1,...,ak with a1,...,ak ∈ A has a solution. Since two congruences θ1, θ2 form a CR tuple if and only if they permute, the property of being a CR tuple is a generalization of the notion of permutability that makes sense for more than two congruences. The main result of this article is a characterization of CR tuples for finite algebras in dual discriminator varieties. As an application, we obtain a neat characterization of CR tuples for finite distributive lattices.