Perfect Hilbert algebras
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RIS BIB ENDNOTEPublication date: 12.12.2024
Reports on Mathematical Logic, 2024, Number 59, pp. 27-48
https://doi.org/10.4467/20842589RM.24.001.20697Authors
Perfect Hilbert algebras
In [S. Celani and L. Cabrer. Duality for finite Hilbert algebras. Discrete Math., 305(1-3):74{99, 2005.] the authors proved that every finite Hilbert algebra A is isomorphic to the Hilbert algebra HK(X) = {w ⇒ i v : w ∈ K and v ⊆ w}, where X is a finite poset, K is a distinguished collection of subsets of X, and the implication ⇒i is defined by: w ⇒i v = {x ∈ X : w ∩ [x) ⊆ v}, where [x) = {y ∈ X : x ≤ y.} The Hilbert implication on HK(X) is the usual Heyting implication ⇒ i (as just defined) given on the increasing subsets. In the same article, Celani and Cabrer extended this representation to a full categorical duality. The aim of the present article is to obtain an algebraic characterization of the Hilbert algebras HK(X) for all structures (X, ≤, K) defined by Celani and Cabrer but not necessarily finite. Then, we shall extend this representation to a full dual equivalence generalizing the finite setting given by Celani and Cabrer.
[1] J. C. Abbott. Semi-boolean algebra. Mat. Vesnik, 4(19):177-198, 1967.
[2] R. A. Bull. Some results for implicational calculi. J. Symb. Logic, 29(1):33-39, 1964.
[3] D. Busneag. A note on deductive systems of a hilbert algebra. Kobe J. Math., 2((1)):29-35, 1985.
[4] J. L. Castiglioni, S. Celani, and H. San Martin. Prelinear Hilbert algebras. Fuzzy Sets and Systems, 397:84-106, 2020.
[5] J. L. Castiglioni, S. A. Celani, and H. J. San Martin. On Hilbert algebras generated by the order. Archive for Mathematical Logic, 61(1):155-172, 2022.
[6] S. Celani. A note on homomorphisms of Hilbert algebras. Int. J. Math. Math. Sci., 29(1):55-61, 2002.
[7] S. Celani. Complete and atomic Tarski algebras. Arch. Math. Logic, 58:1-16, 2019.
[8] S. Celani and L. Cabrer. Duality for finite Hilbert algebras. Discrete Math., 305(1-3):74-99, 2005.
[9] S. Celani, L. Cabrer, and D. Montangie. Representation and duality for Hilbert algebras. Cent. Eur. J. Math., 7(3):463-478, 2009.
[10] I. Chajda and R. Halas. Order algebras. Demonstratio Mathematica, 35(1):1-10, 2002.
[11] A. Diego. Sur les algebres de Hilbert. In Collection de Logique Mathematique, volume 21 of A. Gauthier-Villars, 1966.
[12] M. Gehrke, H. Nagahashi, and Y. Venema. A Sahlqvist theorem for distributive modal logic. Ann. Pure Appl. Logic, 131(1):65-102, 2005.
[13] S. Givant and P. Halmos. Introduction to Boolean algebras. Springer, 2009.
[14] A. Monteiro. Sur les algebres de Heyting symetriques. Portugal. Math., 39(1-4):1-237, 1980.
[15] A. Monteiro. Les Algebres de Hilbert lineaires. In Notas de Logica Matematica, volume 40, pages 1-14. 1996.
[16] H. Rasiowa. An Algebraic Approach to Non-Classical Logics. North-Holland, 1974.
[17] D. P. Smith. Meet-irreducible elements in implicative lattices. Proc. Amer. Math. Soc., 34(1):57-62,1972.
Information: Reports on Mathematical Logic, 2024, Number 59, pp. 27-48
Article type: Original article
CONICET - National Scientific and Technical Research Council
Argentina
National University of La Pampa
Argentina
Published at: 12.12.2024
Received at: 15.12.2023
Article status: Open
Licence: CC BY
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