Congruence systems in dual discriminator varieties

Abstract

A congruence system on an algebra \(\textbf{A}\) is a tuple \(\langle \theta _1,\ldots ,\theta _k,\) \(a_1,\ldots ,a_k\rangle \) where \(\theta _1,\ldots ,\theta _k \in \mathop {\textrm{Con}}\textbf{A}\), \(a_1,\ldots ,a_k \in A\) and \(\langle a_i,a_j\rangle \in \theta _i \vee \theta _j\) for all \(i,j \in \{1,\ldots ,k\}\). A solution to such a congruence system is an element \(a \in A\) satisfying \(\langle a,a_i\rangle \in \theta _i\) for all \(i \in \{1,\ldots ,k\}\). A tuple of congruences \(\langle \theta _1,\ldots , \theta _k\rangle \) is said to be a Chinese Remainder tuple (CR tuple for short) of \(\textbf{A}\) provided that every system \(\langle \theta _1,\ldots ,\theta _k,a_1,\ldots ,a_k\rangle \) with \(a_1,\ldots ,a_k \in A\) has a solution. Since two congruences \(\theta _1,\theta _2\) 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.