Mathematica Bohemica, Vol. 142, No. 3, pp. 225-241, 2017
A topological duality for the $F$-chains associated with the logic $C_\omega$
Verónica Quiroga, Víctor Fernández
Received December 19, 2014. First published December 19, 2016.
Abstract: In this paper we present a topological duality for a certain subclass of the $F_{\omega}$-structures defined by M. M. Fidel, which conform to a non-standard semantics for the paraconsistent N. C. A. da Costa logic $C_\omega$. Actually, the duality introduced here is focused on $F_\omega$-structures whose supports are chains. For our purposes, we characterize every $F_\omega$-chain by means of a new structure that we will call down-covered chain (DCC) here. This characterization will allow us to prove the dual equivalence between the category of $F_\omega$-chains and a new category, whose objects are certain special topological spaces (together with a distinguished family of open sets) and whose morphisms are particular continuous functions.
Keywords: paraconsistent logic; algebraic logic; dualities for ordered structures
References: [1] R. Balbes, P. Dwinger: Distributive Lattices. University of Missouri Press, Columbia (1974). MR 0373985 | Zbl 0321.06012 [2] W. A. Carnielli, J. Marcos: Limits for paraconsistent calculi. Notre Dame J. Formal Logic 40 (1999), 375-390. DOI 10.1305/ndjfl/1022615617 | MR 1845624 | Zbl 1007.03028 [3] S. A. Celani, L. M. Cabrer, D. Montangie: Representation and duality for Hilbert algebras. Cent. Eur. J. Math. 7 (2009), 463-478. DOI 10.2478/s11533-009-0032-5 | MR 2534466 | Zbl 1184.03064 [4] S. A. Celani, D. Montangie: Hilbert algebras with supremum. Algebra Univers. 67 (2012), 237-255. DOI 10.1007/s00012-012-0178-z | MR 2910125 | Zbl 1254.03117 [5] N. C. A. da Costa: On the theory of inconsistent formal systems. Notre Dame J. Formal Logic 15 (1974), 497-510. DOI 10.1305/ndjfl/1093891487 | MR 0354361 | Zbl 0236.02022 [6] B. A. Davey, H. A. Priestley: Introduction to Lattices and Order. Cambridge University Press, Cambridge (1990). MR 1058437 | Zbl 0701.06001 [7] M. M. Fidel: The decidability of the calculi ${\cal C}_n$. Rep. Math. Logic 8 (1977), 31-40. MR 0479957 | Zbl 0378.02011 [8] M. M. Fidel: An algebraic study of logic with constructible negation. Proc. 3rd Brazilian Conf. Mathematical Logic, Recife, 1979 (A. I. Arruda et al., eds.). Soc. Brasil. Lógica, Sao Paulo (1980), 119-129. MR 0603663 | Zbl 0453.03024 [9] R. Jansana, U. Rivieccio: Priestley duality for $N4$-lattices. Proc. 8th Conf. European Society for Fuzzy Logic and Technology, 2013, pp. 263-269. [10] M. Mandelker: Relative annihilators in lattices. Duke Math. J. 37 (1970), 377-386. DOI 10.1215/S0012-7094-70-03748-8 | MR 0256951 | Zbl 0206.29701 [11] S. P. Odintsov: Constructive Negations and Paraconsistency. Trends in Logic - Studia Logica Library 26. Springer, New York (2008). DOI 10.1007/978-1-4020-6867-6 | MR 2680932 | Zbl 1161.03014 [12] H. A. Priestley: Ordered topological spaces and the representation of distributive lattices. Proc. Lond. Math. Soc. (3) 24 (1972), 507-530. DOI 10.1112/plms/s3-24.3.507 | MR 0300949 | Zbl 0323.06011 [13] V. Quiroga: An alternative definition of $F$-structures for the logic $C_1$. Bull. Sect. Log., Univ. Lódź, Dep. Log. 42 (2013), 119-134. MR 3168734 | Zbl 1287.03060 [14] H. Rasiowa, R. Sikorski: The Mathematics of Metamathematics. Monografie Matematyczne 41. Panstwowe Wydawnictwo Naukowe, Warsaw (1963). MR 0163850 | Zbl 0122.24311
Affiliations: Verónica Quiroga, Víctor Fernández, Basic Sciences Institute, National University of San Juan, Av. José Ignacio de la Roza Oeste 230, San Juan 5400, Argentina, e-mail: veronicaandreaquiroga@gmail.com; vlfernan@ffha.unsj.edu.ar
