Abstract
Inspired by an old construction due to J. Kalman that relates distributive lattices and centered Kleene algebras, in this paper we study an equivalence for certain categories whose objects are algebras with implication \({(H, \bigwedge, \bigvee, \rightarrow, 0,1)}\) which satisfy the following property for every \({a,b,c\, \in\, H}\): if \({a \leq b \rightarrow c}\), then \({a \bigwedge b \leq c}\).
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References
Balbes, R., Dwinger P.: Distributive Lattices. University of Missouri Press (1974)
Bezhanishvili N., Gehrke M.: Finitely generated free Heyting algebras via Birkhoff duality and coalgebra. Logical Methods in Computer Science 7, 1–24 (2011)
Busaniche M., Cignoli R.: Constructive Logic with Strong Negation as a Substructural Logic. Journal of Logic and Computation 20, 761–793 (2010)
Castiglioni J.L., Menni M., Sagastume M.: On some categories of involutive centered residuated lattices. Studia Logica 90, 93–124 (2008)
Castiglioni J.L., Lewin R., Sagastume M.: On a definition of a variety of monadic l-groups. Studia Logica 102, 67–92 (2014)
Celani S.: Bounded distributive lattices with fusion and implication. Southeast Asian Bull. Math. 27, 1–10 (2003)
Celani S., Jansana R.: Bounded distributive lattices with strict implication. Mathematical Logic Quarterly 51, 219–246 (2005)
Cignoli R.: The class of Kleene algebras satisfying an interpolation property and Nelson algebras. Algebra Universalis 23, 262–292 (1986)
Epstein G., Horn A.: Logics which are characterized by subresiduated lattices. Z. Math. Logik Grundlagen Math. 22, 199–210 (1976)
Fidel M. M.: An algebraic study of a propositional system of Nelson. In: Arruda, A.I., Da Costa, N.C.A., Chuaqui, R. (eds.) Mathematical Logic. Proceedings of the First Brazilian Conference. Lectures in Pure and Applied Mathematics, vol. 39, pp. 99–117. Marcel Dekker, New York (1978)
Kalman J.A.: Lattices with involution. Trans. Amer. Math. Soc. 87, 485–491 (1958)
Sagastume M.: Kalman’s construction (2007, preprint).
Spinks M., Veroff R.: Constructive logic with strong negation is a substructural logic. I. Studia Logica 88, 325–348 (2008)
Vakarelov D.: Notes on \({\mathcal{N}}\)-lattices and constructive logic with strong negation. Studia Logica 36, 109–125 (1977)
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Presented by C. Tsinakis.
This work was supported by grants CONICET PIP 112-201101-00636 and UNLP 11/X667.
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Castiglioni, J.L., Celani, S.A. & San Martín, H.J. Kleene algebras with implication. Algebra Univers. 77, 375–393 (2017). https://doi.org/10.1007/s00012-017-0433-4
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DOI: https://doi.org/10.1007/s00012-017-0433-4