Congruence Pairs of Algebras Abstracting Double Kleene and Stone algebras

In this note, we extend the result of Beazer on congruence pairs of $K_2$-algebras to the class of double $K_2$-algebras. We show that any congruence on a double $K_2$-algebra can be represented by a congruence pair $\langle θ_1, θ_2\rangle,$ where $θ_1$ is a Kleene congruence and $θ_2$ is a lattice one. As an application of this result, we give a sufficient condition for a double $K_2$-algebra is congruence permutable ($n$-permutable).