On the relation of negations in Nelson algebras

Conrado Gomez,

Miguel Andres Marcos,

Hernan Javier San Martin

Abstrakt

The aim of this paper is to investigate the relation between the strong and the "weak" or intuitionistic negation in Nelson algebras. To do this, we define the variety of Kleene algebras with intuitionistic negation and explore the Kalman's construction for pseudocomplemented distributive lattices. We also study the centered algebras of this variety.

AMS subject classification: 06D16, 03G25.

Słowa kluczowe: negations, A Nelson algebras, pseudocomplemented distributive lattices, Kleene algebras.
References

[1] R. Balbes and P. Dwinger, Distributive Lattices, University of Missouri Press 1974.

[2] D. Brignole and A. Monteiro, Caracterisation des Algebres de Nelson par des Egalites. I, II, Proc. Japan Acad. 43 (1967), 279{285. Reproduced in Notas de Logica Matematica no. 20 (1974), Universidad Nacional del Sur, Baha Blanca.

[3] D. Brignole, Equational Characterization of Nelson Algebras, Notre Dame J. Formal Logic 10 (1969), 285{297. Reproduced in Notas de Logica Matematica no. 9 (1974), Universidad Nacional del Sur, Baha Blanca.

[4] M. Busaniche and R. Cignoli, Constructive Logic with Strong Negation as a Substructural Logic, Journal of Logic and Computation 20:4 (2010), 761-793.

[5] J.L. Castiglioni, S. Celani, and H.J. San Martin, Kleene algebras with implication, Algebra Universalis 77:4 (2017) 375-393.

[6] J.L Castiglioni, R. Lewin, and M. Sagastume, On a definition of a variety of monadic l-groups, Studia Logica 102:1 (2014) 67-92.

[7] J.L. Castiglioni, M. Menni, and M. Sagastume, On some categories of involutive centered residuated lattices, Studia Logica 90:1 (2008), 93-124.

[8] R. Cignoli, The class of Kleene algebras satisfying an interpolation property and Nelson algebras, Algebra Universalis 23 (1986), 262-292.

[9] R. Cignoli, Quantifiers on distributive lattices, Discrete Mathematics 96 (1991), 183-197.

[10] J.M. Cornejo and H.J. San Martin, A categorical equivalence between semi-Heyting algebras and centered semi-Nelson algebras, Logic Journal of the IGPL 26:4 (2018), 408-428.

[11] F. Esteva and X. Domingo, Sobre funciones de negacion en [0; 1], Stochastica 4:2 (1980).

[12] M.M. Fidel, An algebraic study of a propositional system of Nelson, in: Mathematical Logic, Proceedings of the First Brazilian Conference. Arruda A.I., Da Costa N.C.A., Chuaqui R., Editors. Lectures in Pure and Applied Mathematics 39. Marcel Dekker, New York and Basel, 99-117 (1978).

[13] P.R. Halmos, Algebraic Logic, Chelsea Publishing Co. 1962.

[14] R. Jansana and H.J. San Martin, On Kalman's functor for bounded hemi-implicative semilattices and hemi-implicative lattices, Logic Journal of the IGPL 26:1 (2018), 47-82.

[15] J.A. Kalman, Lattices with involution, Trans. Amer. Math. Soc. 87 (1958), 485-491.

[16] L. Monteiro and O. Varsavsky, Algebras de Heyting monadicas, Actas de las X jornadas, Union Matematica Argentina, Instituto de Matematicas, Universidad Nacional del Sur, Baha Blanca, 1957, pp. 52-62 (a French translation is published as Notas de Logica Matematica No 1, Instituto de Matematica, Universidad Nacional del Sur, Baha Blanca, 1974).

[17] D. Nelson, Constructible falsity, J. Symb. Logic 14 (1949), 16-26.

[18] A. Petrovich, Monadic De Morgan algebras, in: Models, Algebras, and Proofs. X. Caicedo and C.H. Montenegro, Editors, Lecture Notes in Pure and Applied Mathematics 203, 315-333 (1999).

[19] M.S. Rao and K. Shum, Boolean  filters of distributive lattices, International Journal of Mathematics and Soft Computing 3:3 (2013) 41-48.

[20] H. Rasiowa, N-lattices and constructive logic with strong negation, Fund. Math. 46 (1958), 61-80.

[21] H. Rasiowa, An algebraic approach to non-classical logics, Studies in logic and the Foundations of Mathematics 78, North-Holland and PNN 1974.

[22] M. Sagastume and H.J. San Martin, The logic L•, Mathematical Logic Quarterly 60:6 (2014), 375-388.

[23] M. Sagastume and H.J. San Martin, A Categorical Equivalence Motivated by Kalman's Construction, Studia Logica 104:2 (2016), 185-206.

[24] A. Sendlewski, Nelson algebras through Heyting ones: I, Studia Logica 49 (1990), 105-126.

[25] A. Sendlewski, Topologicality of Kleene Algebras With a Weak Pseudocomplementation Over Distributive P-Algebras, Reports on Mathematical Logic 25 (1991).

[26] M. Spinks and R. Veroff, Constructive logic with strong negation is a substructural logic. I, Studia Logica 88 (2008), 325-348.

[27] M. Spinks and R. Veroff, Constructive logic with strong negation is a substructural logic. II, Studia Logica 89 (2008), 401-425.

[28] D. Vakarelov, Notes on N-lattices and constructive logic with strong negation, Studia Logica 34 (1977), 109-125.

[29] D. Vakarelov, Nelson's Negation on the base of Weaker Versions of Intuitionistic Negation, Studia Logica 80 (2005), 393-430.

[30] I. Viglizzo, Algebras de Nelson, Tesis de Magister, Universidad Nacional del Sur, Baha Blanca, Buenos Aires 1999.