DISCRETE DUALITY FOR TSH-ALGEBRAS

Title & Authors
DISCRETE DUALITY FOR TSH-ALGEBRAS
Figallo, Aldo Victorio; Pelaitay, Gustavo; Sanza, Claudia;

Abstract
In this article, we continue the study of tense symmetric Heyting algebras (or TSH-algebras). These algebras constitute a generalization of tense algebras. In particular, we describe a discrete duality for TSH-algebras bearing in mind the results indicated by Or lowska and Rewitzky in [E. Orlowska and I. Rewitzky, Discrete Dualities for Heyting Algebras with Operators, Fund. Inform. 81 (2007), no. 1-3, 275-295] for Heyting algebras. In addition, we introduce a propositional calculus and prove this calculus has TSH-algebras as algebraic counterpart. Finally, the duality mentioned above allowed us to show the completeness theorem for this calculus.
Keywords
symmetric Heyting algebras;tense operators;frames;discrete duality;
Language
English
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References
1.
R. Balbes and P. Dwinger, Distributive Lattices, University of Missouri Press, Columbia, Miss, 1974.

2.
M. Botur, I. Chajda, R. Halas, and M. Kolarik, Tense operators on basic algebras, to appear in Internat. J. of Theoret. Phys.

3.
J. Burgess, Basic tense logic, Handbook of philosophical logic, Vol. II, 89-133, Synthese Lib., 165, Reidel, Dordrecht, 1984.

4.
I. Chajda, Algebraic axiomatization of tense intuitionistic logic, Cent. Eur. J. Math. 9 (2011), no. 5, 1185-1191.

5.
C. Chirita, Tense $\theta$-valued Moisil propositional logic, Int. J. of Computers, Communications & Control 5 (2010), no. 5, 642-653.

6.
C. Chirita, Tense $\theta$-valued Lukasiewicz-Moisil algebras, Journal of Multiple Valued Logic and Soft Computing 17 (2011), no. 1, 1-24.

7.
D. Diaconescu and G. Georgescu, Tense operators on MV -algebras and Lukasiewicz-Moisil algebras, Fund. Inform. 81 (2007), no. 4, 379-408.

8.
W. Dzik, E. Orlowska, and C. van Alten, Relational representation theorems for general lattices with negations, Relations and Kleene algebra in computer science, 162-176, Lecture Notes in Comput. Sci., 4136, Springer, Berlin, 2006.

9.
A. V. Figallo, C. Gallardo, and G. Pelaitay, Tense operators on m-symmetric algebras, Int. Math. Forum 41 (2011), no. 6, 2007-2014.

10.
A. V. Figallo and G. Pelaitay, Tense operators on SHn-algebras, Pioneer Journal of Algebra, Number Theory and its Aplications 1 (2011), no. 1, 33-41.

11.
A. V. Figallo, G. Pelaitay, and C. Sanza, Operadores temporales sobre algebras de Heyting simetricas, LIX Reunion Anual de la Union Matematica Argentina. Universidad Nacional de Mar del Plata, 2009.

12.
L. Iturrioz, Symmetrical Heyting algebras with operators, Z. Math. Logik Grundlag. Math. 29 (1983), no. 1, 33-70.

13.
T. Kowalski, Varieties of tense algebras, Rep. Math. Logic 32 (1998), 53-95.

14.
Gr. C. Moisil, Logique modale, Disquisit. Math. Phys. 2 (1942), 3-98.

15.
A. Monteiro, Sur les algebres de Heyting Simetriques, Special issue in honor of Antonio Monteiro, Portugal. Math. 39 (1980), no. 1-4, 1-237.

16.
E. Or lowska and I. Rewitzky, Discrete Dualities for Heyting Algebras with Operators, Fund. Inform. 81 (2007), no. 1-3, 275-295.

17.
E. Or lowska and I. Rewitzky, Duality via Truth: Semantic frameworks for lattice-based logics, Log. J. IGPL 13 (2005), no. 4, 467-490.

18.
H. Rasiowa, An Algebraic Approach to Non-Classical Logics, Studies in Logic and the Foundations of Mathematics, 78, North-Holland, 1974.

19.
H. P. Sankappanavar, Heyting algebras with a dual lattice endomorphism, Z. Math. Logik Grundlag. Math. 33 (1987), no. 6, 565-573.