Congruence relations in direct products

This paper studies conditions under which every congruence relation $\theta$ in a direct product $A \times B$ of algebras is of the form ${\theta _1} \times {\theta _2}$, where ${\theta _1}$ and ${\theta _2}$ are congruence relations in $A$ and $B$ respectively. It is shown that for any equational class $K$, every such $\theta$ in every $A \times B$ in $K$ has this property if and only if $K$ satisfies certain identities.