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Regular Double p-Algebras: A Converse to a Katriňák Theorem and Applications

  • Juan M. Cornejo , Michael Kinyon and Hanamantagouda P. Sankappanavar EMAIL logo
Published/Copyright: December 18, 2023
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 73 Issue 6

ABSTRACT

In 1973, Katriňák proved that regular double p-algebras can be regarded as (regular) double Heyting algebras by ingeniously constructing binary terms for the Heyting implication and its dual in terms of pseudocomplement and its dual. In this paper, we prove a converse to Katriňák’s theorem, in the sense that in the variety ℝDℙℂℍ of regular dually pseudocomplemented Heyting algebras, the implication operation → satisfies Katriňák’s formula. As applications of this result together with the above-mentioned Katriňák’s theorem, we show that the varieties ℝDBLℙ , ℝDℙℂℍ , ℝℙℂℍd and ℝDBLℍ of regular double p-algebras, regular dually pseudocomplemented Heyting algebras, regular pseudocomplemented dual Heyting algebras, and regular double Heyting algebras, respectively, are term-equivalent to each other and also that the varieties ℝDMℙ , ℝDMℍ , ℝDMDBLℍ , ℝDMDBLℙ of regular De Morgan p-algebras, regular De Morgan Heyting algebras, regular De Morgan double Heyting algebras, and regular De Morgan double p-algebras, respectively, are also term-equivalent to each other. From these results and recent results of Adams, Sankappanavar and Vaz de Carvalho on varieties of regular double p-algebras and regular pseudocomplemented De Morgan algebras, we deduce that the lattices of subvarieties of all these varieties have cardinality 2ℵ0 . We then define new logics, ℛDPCℋ , ℛPCℋd and ℛDℳℋ , and show that they are algebraizable with ℝDℙℂℍ , ℝℙℂℍd and ℝDMℍ , respectively, as their equivalent algebraic semantics. It is also deduced that the lattices of extensions of all of the above mentioned logics have cardinality 2ℵ0 .

2020 Mathematics Subject Classification: Primary 03G25; 06D20; 08B15; 06D15; 03C05; 03B50; Secondary 08B26; 06D30; 06E75
Keywords: Regular double p-algebra; regular dually pseudocomplemented Heyting algebra; regular pseudocomplemented dual Heyting algebras; and regular double Heyting algebra; regular De Morgan p-algebras; regular De Morgan Heyting algebras; regular De Morgan double Heyting algebras; regular De Morgan double p-algebras; logic RDPCH; logic RPCHd; logic RDMH

(Communicated by Anatolij Dvurečenskij)

In Memory of Professor Tibor Katriňák

Funding statement: The first author wants to thank the institutional support of CONICET (Consejo Nacional de Investigaciones Científicas y Técnicas) and Universidad Nacional del Sur.

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Received: 2022-10-19
Accepted: 2022-12-21
Published Online: 2023-12-18

© 2023 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2023-0099
Keywords for this article
Regular double p-algebra; regular dually pseudocomplemented Heyting algebra; regular pseudocomplemented dual Heyting algebras; and regular double Heyting algebra; regular De Morgan p-algebras; regular De Morgan Heyting algebras; regular De Morgan double Heyting algebras; regular De Morgan double p-algebras; logic RDPCH; logic RPCHd; logic RDMH

Articles in the same Issue

  1. ℒ-fuzzy Annihilators in Residuated Lattices
  2. Regular Double p-Algebras: A Converse to a Katriňák Theorem and Applications
  3. Uniform Local Connectedness and Completion of Metric σ-Frames
  4. On Transcendental Regular Continued Fractions
  5. On Repdigits Which are Sums or Differences of Two k-Pell Numbers
  6. Peirce Decompositions for Evolution Algebras
  7. Generalized Anti-Symmetry Laws in Groupoids
  8. New Bounds of Cyclic Jensen’s Differences via Weighted Hadamard Inequalities with Applications
  9. Geometric Properties of Generalized Bessel Function Associated with the Exponential Function
  10. On the Attainable Set of Iterative Differential Inclusions
  11. On the Differential-Difference Sine-Gordon Equation with an Integral Type Source
  12. A Critical p(x)-Laplacian Steklov Type Problem with Weights
  13. Distributivity of a Uni-nullnorm with Continuous and Archimedean Underlying T-norms and T-conorms Over an Arbitrary Uninorm
  14. Some Approximation Properties of Operators Including Degenerate Appell Polynomials
  15. Operators that are Weakly Coercive and a Compact Perturbation
  16. On the Structure Lie Operator of a Real Hypersurface in the Complex Quadric
  17. Paracompactness and Open Relations
  18. On (Strongly) (Θ-)Discrete Homogeneous Spaces
  19. The Alpha Power Muth-G Distributions and Its Applications in Survival and Reliability Analyses
  20. Weakly Einstein Equivalence in a Golden Space Form and Certain CR-submanifolds
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