A new categorical equivalence for stone algebras
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Ismael Calomino
und
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.Suche in Google Scholar
[2] Brignole, D.: Equational characterization of Nelson Algebras, Notre Dame J. Formal Logic 10 (1969), 285–297.10.1305/ndjfl/1093893718Suche 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/1195521625Suche in Google Scholar
[4] Celani, S. A.: Classical modal De Morgan algebras, Stud. Log. 98 (2011), 251–266.10.1007/s11225-011-9328-0Suche 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-xSuche in Google Scholar
[6] Chen, C. C.—Grätzer, G.: Stone lattices. I: Construction theorems, Canad. J. Math. 21 (1969), 884–894.10.4153/CJM-1969-096-5Suche in Google Scholar
[7] Chen, C. C.—Grätzer, G.: Stone lattices. II: Structure theorems, Canad. J. Math. 21 (1969), 895–903.10.4153/CJM-1969-097-2Suche 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/BF01230621Suche in Google Scholar
[9] Düntsch, I.: A logic for rough sets, Theor. Comput. Sci. 179 (1997), 427–436.10.1016/S0304-3975(96)00334-9Suche 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.Suche in Google Scholar
[11] Gehrke, M.—Walker, E.: On the structure of rough sets, Bull. Pol. Acad. Sci. Math. 40 (1992), 235–245.Suche 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.21.002.14374Suche in Google Scholar
[13] Grätzer, G.: Lattice Theory: Foundation, Birkhäuser Springer Basel, Basel, 2011.10.1007/978-3-0348-0018-1Suche 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/BF02020328Suche 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/S1793557119500104Suche 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-2014-1010Suche in Google Scholar
[17] Kalman, J. A.: Lattices with involution, Trans. Amer. Math. Soc. 87 (1958), 485–491.10.1090/S0002-9947-1958-0095135-XSuche 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/S0002-9939-1973-0316335-0Suche in Google Scholar
[19] Katriňák, T.—Žabka, M.: A weak Boolean representation of double Stone algebras, Houston J. Math. 30 (2004), 615–628.Suche 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.2024.109309Suche in Google Scholar
[21] Pelaitay, G.—Starobinsky, M.: A categorical equivalence for tense pseudocomplemented distributive lattices, J. Appl. Log. 11 (2024), 237–251.Suche in Google Scholar
[22] Pomykała, J.—Pomykała, J. A.: The Stone algebra of rough sets, Bull. Pol. Acad. Sci. Math. 36 (1988), 495–508.Suche in Google Scholar
[23] Rasiowa, H.: N-lattices and constructive logic with strong negation, Fund. Math. 46 (1958), 61–80.10.4064/fm-46-1-61-80Suche in Google Scholar
[24] Sendlewski, A.: Nelson algebras through Heyting ones. I, Stud. Log. 49 (1990), 105–126.10.1007/BF00401557Suche 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.Suche in Google Scholar
[26] Sholander, M.: Postulates for distributive lattices, Canad. J. Math. 3 (1951), 28–30.10.4153/CJM-1951-003-5Suche in Google Scholar
[27] Vakarelov, D.: Notes on N-lattices and constructive logic with strong negation, Stud. Log. 36 (1977), 109–125.10.1007/BF02121118Suche in Google Scholar
[28] Walker, E.A.: Stone algebras, conditional events, and three-valued logic, IEEE Trans. Syst. Man Cybern. 24 (1994), 1699–1707.10.1109/21.328927Suche in Google Scholar
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Artikel in diesem Heft
- 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
