Permutabitity of principal $ MS $-algebras

Abd El-Mohsen Badawy; Alaa Helmy; Abd El-Mohsen Badawy; Alaa Helmy

doi:10.3934/math.20231012

AIMS Mathematics

2023, Volume 8, Issue 9: 19857-19875. doi: 10.3934/math.20231012

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Research article

Permutabitity of principal $ MS $-algebras

Abd El-Mohsen Badawy ¹,
Alaa Helmy ^{2
,
,}

1.
Department of Mathematics Faculty of Science, Tanta University, Egypt
2.
Department of Mathematics Faculty of Science, Kafrelsheikh University, Egypt

Received: 31 March 2023 Revised: 31 May 2023 Accepted: 05 June 2023 Published: 14 June 2023
MSC : 06D05, 06D30

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 Con_{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 $ \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.
- $ MS $-algebras,
- principal $ MS $-algebras,
- congruence relation,
- $ MS $-congruence pairs,
- $ 2 $-permutability of congruences,
- $ n $-permutability of congruences
Citation: Abd El-Mohsen Badawy, Alaa Helmy. Permutabitity of principal $ MS $-algebras[J]. AIMS Mathematics, 2023, 8(9): 19857-19875. doi: 10.3934/math.20231012

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Abstract

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 Con_{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 $ \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.

References

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