A new categorical equivalence for stone algebras
-
Ismael Calomino
and
Gustavo Pelaitay
Abstract
The aim of this paper is to give a categorical equivalence for Stone algebras. We introduce the variety of Stone-Kleene algebras with intuitionistic negation, or Stone KAN-algebras for short, and explore Kalman’s construction for Stone algebras. We examine the centered algebras within this new variety and prove that the category of Stone algebras is equivalent to the category of centered Stone KAN-algebras. Moreover, inspired by Monteiro’s construction for Nelson algebras, we propose a method to construct a centered Stone KAN-algebra from a given Stone KAN-algebra and show the connection between Kalman’s construction and Monteiro’s construction.
(Communicated by Roberto Giuntini)
References
[1] Balbes, R.—Dwinger, P.: Distributive Lattices, University of Missouri Press, Columbia, 1974.Search in Google Scholar
[2] Brignole, D.: Equational characterization of Nelson Algebras, Notre Dame J. Formal Logic 10 (1969), 285–297.10.1305/ndjfl/1093893718Search in Google Scholar
[3] Brignole, D.—Monteiro, A.: Caractrisation des algèbres de Nelson par des ègalitès. I, II, Proc. Japan Acad. 43 (1967), 284–285.10.3792/pja/1195521625Search in Google Scholar
[4] Celani, S. A.: Classical modal De Morgan algebras, Stud. Log. 98 (2011), 251–266.10.1007/s11225-011-9328-0Search in Google Scholar
[5] Chen, L.—Shi, N.—Wu, G.: Definable sets in Stone algebras, Arch. Math. Logic 55 (2016), 749–757.10.1007/s00153-016-0491-xSearch in Google Scholar
[6] Chen, C. C.—Grätzer, G.: Stone lattices. I: Construction theorems, Canad. J. Math. 21 (1969), 884–894.10.4153/CJM-1969-096-5Search in Google Scholar
[7] Chen, C. C.—Grätzer, G.: Stone lattices. II: Structure theorems, Canad. J. Math. 21 (1969), 895–903.10.4153/CJM-1969-097-2Search in Google Scholar
[8] Cignoli, R.: The class of Kleene algebras satisfying an interpolation property and Nelson algebras, Algebra Universalis 23 (1986), 262–292.10.1007/BF01230621Search in Google Scholar
[9] Düntsch, I.: A logic for rough sets, Theor. Comput. Sci. 179 (1997), 427–436.10.1016/S0304-3975(96)00334-9Search in Google Scholar
[10] Fidel, M. M.: An algebraic study of a propositional system of Nelson, Math. Logic, Proc. 1st Braz. Conf., Campinas 1977, Lect. Notes Pure Appl. Math. 39 (1978), 99–117.Search in Google Scholar
[11] Gehrke, M.—Walker, E.: On the structure of rough sets, Bull. Pol. Acad. Sci. Math. 40 (1992), 235–245.Search in Google Scholar
[12] Gomez, C.—Marcos, M.—San Martín, H. J.: On the relation of negations in Nelson algebras, Rep. Math. Logic 56 (2021), 15–56.10.4467/20842589RM.21.002.14374Search in Google Scholar
[13] Grätzer, G.: Lattice Theory: Foundation, Birkhäuser Springer Basel, Basel, 2011.10.1007/978-3-0348-0018-1Search in Google Scholar
[14] Grätzer, G.—Schmidt, E. T.: On a problem of M. H. Stone, Acta Math. Acad. Sci. Hung. 8 (1957), 455–460.10.1007/BF02020328Search in Google Scholar
[15] Gregori, V.: Discrete duality for De Morgan algebras with operators, Asian-Eur. J. Math. 12 (2019), Art. ID 1950010.10.1142/S1793557119500104Search in Google Scholar
[16] Järvinen, J.—Radeleczki, S.: Monteiro spaces and rough sets determined by quasiorder relations: models for Nelson algebras, Fund. Inform. 131 (2014), 205–215.10.3233/FI-2014-1010Search in Google Scholar
[17] Kalman, J. A.: Lattices with involution, Trans. Amer. Math. Soc. 87 (1958), 485–491.10.1090/S0002-9947-1958-0095135-XSearch in Google Scholar
[18] Katriňák, T.: A new proof of the construction theorem for Stone algebras, Proc. Amer. Math. Soc. 40 (1973), 75–78.10.1090/S0002-9939-1973-0316335-0Search in Google Scholar
[19] Katriňák, T.—Žabka, M.: A weak Boolean representation of double Stone algebras, Houston J. Math. 30 (2004), 615–628.Search in Google Scholar
[20] Kumar, A.—Gaur, N.—Dewan, B.: A 4-valued logic for double Stone algebras, Int. J. Approx. Reasoning 176 (2024), Art. ID 109309.10.1016/j.ijar.2024.109309Search in Google Scholar
[21] Pelaitay, G.—Starobinsky, M.: A categorical equivalence for tense pseudocomplemented distributive lattices, J. Appl. Log. 11 (2024), 237–251.Search in Google Scholar
[22] Pomykała, J.—Pomykała, J. A.: The Stone algebra of rough sets, Bull. Pol. Acad. Sci. Math. 36 (1988), 495–508.Search in Google Scholar
[23] Rasiowa, H.: N-lattices and constructive logic with strong negation, Fund. Math. 46 (1958), 61–80.10.4064/fm-46-1-61-80Search in Google Scholar
[24] Sendlewski, A.: Nelson algebras through Heyting ones. I, Stud. Log. 49 (1990), 105–126.10.1007/BF00401557Search in Google Scholar
[25] Sendlewski, A.: Topologicality of Kleene algebras with a weak pseudocomplementation over distributive p-algebras, Rep. Math. Logic 25 (1991), 13–56.Search in Google Scholar
[26] Sholander, M.: Postulates for distributive lattices, Canad. J. Math. 3 (1951), 28–30.10.4153/CJM-1951-003-5Search in Google Scholar
[27] Vakarelov, D.: Notes on N-lattices and constructive logic with strong negation, Stud. Log. 36 (1977), 109–125.10.1007/BF02121118Search in Google Scholar
[28] Walker, E.A.: Stone algebras, conditional events, and three-valued logic, IEEE Trans. Syst. Man Cybern. 24 (1994), 1699–1707.10.1109/21.328927Search in Google Scholar
© 2025 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- A new categorical equivalence for stone algebras
- On special classes of prime filters in BL-algebras
- A note on characterized and statistically characterized subgroups of 𝕋 = ℝ/ℤ
- New Young-type integral inequalities using composition schemes
- The structure of pseudo-n-uninorms with continuous underlying functions
- Jensen-type inequalities for a second-order differential inequality condition
- A direct proof of the characterization of the convexity of the discrete Choquet integral
- Envelope of plurifinely plurisubharmonic functions and complex Monge-Ampère type equation
- Fekete-Szegö inequalities for Φ-parametric and β-spirllike mappings of complex order in ℂn
- Entire function sharing two values partially with its derivative and a conjecture of Li and Yang
- Oscillatory properties of third-order semi-canonical dynamic equations on time scales via canonical transformation
- Weighted B-summability and positive linear operators
- Some properties and applications of convolution algebras
- On measures of σ-noncompactess in F-spaces
- On the kolmogorov–feller–gut weak law of large numbers for triangular arrays of rowwise and pairwise negatively dependent random variables
- Intermediately trimmed sums of oppenheim expansions: A strong law
- Novel weighted distribution: Properties, applications and web-tool
- On the q-Gamma distribution: Properties and inference
- Finiteorthoatomistic effect algebras and regular algebraic E-test spaces
- Prof. RNDr. Anatolij Dvurečenskij, DrSc. 75th anniversary
Articles in the same Issue
- A new categorical equivalence for stone algebras
- On special classes of prime filters in BL-algebras
- A note on characterized and statistically characterized subgroups of 𝕋 = ℝ/ℤ
- New Young-type integral inequalities using composition schemes
- The structure of pseudo-n-uninorms with continuous underlying functions
- Jensen-type inequalities for a second-order differential inequality condition
- A direct proof of the characterization of the convexity of the discrete Choquet integral
- Envelope of plurifinely plurisubharmonic functions and complex Monge-Ampère type equation
- Fekete-Szegö inequalities for Φ-parametric and β-spirllike mappings of complex order in ℂn
- Entire function sharing two values partially with its derivative and a conjecture of Li and Yang
- Oscillatory properties of third-order semi-canonical dynamic equations on time scales via canonical transformation
- Weighted B-summability and positive linear operators
- Some properties and applications of convolution algebras
- On measures of σ-noncompactess in F-spaces
- On the kolmogorov–feller–gut weak law of large numbers for triangular arrays of rowwise and pairwise negatively dependent random variables
- Intermediately trimmed sums of oppenheim expansions: A strong law
- Novel weighted distribution: Properties, applications and web-tool
- On the q-Gamma distribution: Properties and inference
- Finiteorthoatomistic effect algebras and regular algebraic E-test spaces
- Prof. RNDr. Anatolij Dvurečenskij, DrSc. 75th anniversary
