Geodesic
A topological duality for the $F$-chains associated with the logic $C_\omega $
Mathematica Bohemica, Tome 142 (2017) no. 3, pp. 225-241
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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 $-structures whose supports are chains. For our purposes, we characterize every \mbox {$F_\omega $-chain} by means of a new structure that we will call {\it 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.
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 $-structures whose supports are chains. For our purposes, we characterize every \mbox {$F_\omega $-chain} by means of a new structure that we will call {\it 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.
DOI : 10.21136/MB.2016.0079-14
Classification : 03B53, 03G10, 06D50
Keywords: paraconsistent logic; algebraic logic; dualities for ordered structures 
@article{10_21136_MB_2016_0079_14,
     author = {Quiroga, Ver\'onica and Fern\'andez, V{\'\i}ctor},
     title = {A topological duality for the $F$-chains associated with the logic $C_\omega $},
     journal = {Mathematica Bohemica},
     pages = {225--241},
     year = {2017},
     volume = {142},
     number = {3},
     doi = {10.21136/MB.2016.0079-14},
     mrnumber = {3695464},
     zbl = {06770143},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.2016.0079-14/}
}
TY  - JOUR
AU  - Quiroga, Verónica
AU  - Fernández, Víctor
TI  - A topological duality for the $F$-chains associated with the logic $C_\omega $
JO  - Mathematica Bohemica
PY  - 2017
SP  - 225
EP  - 241
VL  - 142
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.21136/MB.2016.0079-14/
DO  - 10.21136/MB.2016.0079-14
LA  - en
ID  - 10_21136_MB_2016_0079_14
ER  - 
%0 Journal Article
%A Quiroga, Verónica
%A Fernández, Víctor
%T A topological duality for the $F$-chains associated with the logic $C_\omega $
%J Mathematica Bohemica
%D 2017
%P 225-241
%V 142
%N 3
%U http://geodesic.mathdoc.fr/articles/10.21136/MB.2016.0079-14/
%R 10.21136/MB.2016.0079-14
%G en
%F 10_21136_MB_2016_0079_14
Quiroga, Verónica; Fernández, Víctor. A topological duality for the $F$-chains associated with the logic $C_\omega $. Mathematica Bohemica, Tome 142 (2017) no. 3, pp. 225-241. doi: 10.21136/MB.2016.0079-14

[1] Balbes, R., Dwinger, P.: Distributive Lattices. University of Missouri Press, Columbia (1974). | MR | JFM

[2] Carnielli, W. A., Marcos, J.: Limits for paraconsistent calculi. Notre Dame J. Formal Logic 40 (1999), 375-390. | DOI | MR | JFM

[3] Celani, S. A., Cabrer, L. M., Montangie, D.: Representation and duality for Hilbert algebras. Cent. Eur. J. Math. 7 (2009), 463-478. | DOI | MR | JFM

[4] Celani, S. A., Montangie, D.: Hilbert algebras with supremum. Algebra Univers. 67 (2012), 237-255. | DOI | MR | JFM

[5] Costa, N. C. A. da: On the theory of inconsistent formal systems. Notre Dame J. Formal Logic 15 (1974), 497-510. | DOI | MR | JFM

[6] Davey, B. A., Priestley, H. A.: Introduction to Lattices and Order. Cambridge University Press, Cambridge (1990). | MR | JFM

[7] Fidel, M. M.: The decidability of the calculi ${\cal C}_n$. Rep. Math. Logic 8 (1977), 31-40. | MR | JFM

[8] Fidel, M. M.: 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 | JFM

[9] Jansana, R., Rivieccio, U.: Priestley duality for $N4$-lattices. Proc. 8th Conf. European Society for Fuzzy Logic and Technology, 2013, pp. 263-269.

[10] Mandelker, M.: Relative annihilators in lattices. Duke Math. J. 37 (1970), 377-386. | DOI | MR | JFM

[11] Odintsov, S. P.: Constructive Negations and Paraconsistency. Trends in Logic---Studia Logica Library 26. Springer, New York (2008). | DOI | MR | JFM

[12] Priestley, H. A.: Ordered topological spaces and the representation of distributive lattices. Proc. Lond. Math. Soc. (3) 24 (1972), 507-530. | DOI | MR | JFM

[13] Quiroga, V.: An alternative definition of $F$-structures for the logic $C_1$. Bull. Sect. Log., Univ. Łódź, Dep. Log. 42 (2013), 119-134. | MR | JFM

[14] Rasiowa, H., Sikorski, R.: The Mathematics of Metamathematics. Monografie Matematyczne 41. Panstwowe Wydawnictwo Naukowe, Warsaw (1963). | MR | JFM

Cité par Sources :