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A topological duality for the \(F\)-chains associated with the logic \(C_\omega\). (English) Zbl 1463.06058

Summary: In this paper we present a topological duality for a certain subclass of the \(F_\omega\)-structures defined by M. M. Fidel [Rep. Math. Logic 8 (1977), 31–40 (1978; Zbl 0378.02011)], which conform to a non-standard semantics for the paraconsistent da Costa logic \(C_\omega\) [N. C. A. da Costa, Notre Dame J. Formal Logic 15, 497–510 (1974; Zbl 0236.02022)]. 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.

MSC:

06D50 Lattices and duality
03G10 Logical aspects of lattices and related structures
03B53 Paraconsistent logics
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