Permutabitity of principal $MS$-algebras

In this paper, we continue to introduce new properties of principal $MS$-algebras deal with congruence relations via $MS$-congruence pairs. Necessary and sufficient conditions for a pair of congruences $({\theta }_1,{\theta }_2)\in Con(L^{\circ \circ })\times Co{n}_{lat}(D(L))$ to become an $MS$-congruence pair of a principal $MS$-algebra (principal Stone algebra) $L$ are obtained. We describe the lattice of all $MS$-congruence pairs of a principal $MS$-algebra $L$ which induced by the Boolean elements of $L$. We introduce certain special congruence $\mathrm{\Psi }$ on a principal $MS$-algebra and its related properties which are useful for the topic of this paper. A characterization of $2$-permutable congruences using $MS$-congruence pairs of principal $MS$-algebras is established. Finally, a characterization of $n$-permutability of congruences of principal $MS$-algebras is given, which is a generalization of the characterization of $2$-permutability of congruences of such algebras.