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Congruence pairs of principal MS-algebras and perfect extensions

  • Abd El-Mohsen Badawy , Miroslav Haviar EMAIL logo and Miroslav Ploščica
Published/Copyright: December 10, 2020
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 70 Issue 6

Abstract

The notion of a congruence pair for principal MS-algebras, simpler than the one given by Beazer for K2-algebras [6], is introduced. It is proved that the congruences of the principal MS-algebras L correspond to the MS-congruence pairs on simpler substructures L°° and D(L) of L that were associated to L in [4].

An analogy of a well-known Grätzer’s problem [11: Problem 57] formulated for distributive p-algebras, which asks for a characterization of the congruence lattices in terms of the congruence pairs, is presented here for the principal MS-algebras (Problem 1). Unlike a recent solution to such a problem for the principal p-algebras in [2], it is demonstrated here on the class of principal MS-algebras, that a possible solution to the problem, though not very descriptive, can be simple and elegant.

As a step to a more descriptive solution of Problem 1, a special case is then considered when a principal MS-algebra L is a perfect extension of its greatest Stone subalgebra LS. It is shown that this is exactly when de Morgan subalgebra L°° of L is a perfect extension of the Boolean algebra B(L). Two examples illustrating when this special case happens and when it does not are presented.

Keywords: principal MS-algebra; MS-congruence pair
MSC 2010: Primary 06B10; Secondary 06D15

The second author acknowledges support from Slovak grant VEGA 1/0337/16. The third author has been supported by VEGA grant 1/0097/18

†

See Remark 1.

  1. Dedicated to the memory of Professor Beloslav Riečan†

  2. Communicated by Anatolij Dvurečenskij

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Received: 2019-12-20
Accepted: 2020-03-03
Published Online: 2020-12-10
Published in Print: 2020-12-16

© 2020 Mathematical Institute Slovak Academy of Sciences

Articles in the same Issue

  1. Regular papers
  2. Fuzzy deductive systems of RM algebras
  3. Congruence pairs of principal MS-algebras and perfect extensions
  4. The lattices of 𝔏-fuzzy state filters in state residuated lattices
  5. Central lifting property for orthomodular lattices
  6. EQ-Modules
  7. On the exponential Diophantine equation Pxn + Pxn+1 + ⋯ + Pxn+k-1 = Pm
  8. Remarks on some generalization of the notion of microscopic sets
  9. Disjointness of composition operators on Hv0 spaces
  10. A common fixed point theorem for non-self mappings in strictly convex menger PM-spaces
  11. The Poincaré-Cartan forms of one-dimensional variational integrals
  12. Coarse cohomology with twisted coefficients
  13. Divisible extension of probability
  14. Asymptotic behavior of the records of multivariate random sequences in a norm sense
  15. Strong convergence of the functional nonparametric relative error regression estimator under right censoring
  16. A new kumaraswamy generalized family of distributions: Properties and applications
  17. Efficient message transmission via twisted Edwards curves
  18. Computation of several Hessenberg determinants
https://doi.org/10.1515/ms-2017-0430
Keywords for this article
principal MS-algebra; MS-congruence pair

Articles in the same Issue

  1. Regular papers
  2. Fuzzy deductive systems of RM algebras
  3. Congruence pairs of principal MS-algebras and perfect extensions
  4. The lattices of 𝔏-fuzzy state filters in state residuated lattices
  5. Central lifting property for orthomodular lattices
  6. EQ-Modules
  7. On the exponential Diophantine equation Pxn + Pxn+1 + ⋯ + Pxn+k-1 = Pm
  8. Remarks on some generalization of the notion of microscopic sets
  9. Disjointness of composition operators on Hv0 spaces
  10. A common fixed point theorem for non-self mappings in strictly convex menger PM-spaces
  11. The Poincaré-Cartan forms of one-dimensional variational integrals
  12. Coarse cohomology with twisted coefficients
  13. Divisible extension of probability
  14. Asymptotic behavior of the records of multivariate random sequences in a norm sense
  15. Strong convergence of the functional nonparametric relative error regression estimator under right censoring
  16. A new kumaraswamy generalized family of distributions: Properties and applications
  17. Efficient message transmission via twisted Edwards curves
  18. Computation of several Hessenberg determinants
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