@article{Badawy2023, 
author = {Abd El-Mohsen Badawy and Alaa Helmy},
title = {Permutabitity of principal    M  S-algebras},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {9},
pages = {19857-19875},
keywords = {MS-algebras, principal MS-algebras, congruence relation, MS-congruence pairs, 2-permutability of congruences, n-permutability of congruences},
url = {https://www.sciopen.com/article/10.3934/math.20231012},
doi = {10.3934/math.20231012},
abstract = {In this paper, we continue to introduce new properties of principal    M  S-algebras deal with congruence relations via    M  S-congruence pairs. Necessary and sufficient conditions for a pair of congruences    (      θ          1        ,      θ          2        )  ∈  C  o  n  (      L          ∘      ∘        )  ×  C  o      n          l      a      t        (  D  (  L  )  ) to become an    M  S-congruence pair of a principal    M  S-algebra (principal Stone algebra)    L are obtained. We describe the lattice of all    M  S-congruence pairs of a principal    M  S-algebra    L which induced by the Boolean elements of    L. We introduce certain special congruence    Ψ on a principal    M  S-algebra and its related properties which are useful for the topic of this paper. A characterization of    2-permutable congruences using    M  S-congruence pairs of principal    M  S-algebras is established. Finally, a characterization of    n-permutability of congruences of principal    M  S-algebras is given, which is a generalization of the characterization of    2-permutability of congruences of such algebras.}
}