Fuzzy Topology and Łukasiewicz Logics from the Viewpoint of Duality Theory

Abstract

This paper explores relationships between many-valued logic and fuzzy topology from the viewpoint of duality theory. We first show a fuzzy topological duality for the algebras of Lukasiewicz n-valued logic with truth constants, which generalizes Stone duality for Boolean algebras to the n-valued case via fuzzy topology. Then, based on this duality, we show a fuzzy topological duality for the algebras of modal Lukasiewicz n-valued logic with truth constants, which generalizes Jónsson-Tarski duality for modal algebras to the n-valued case via fuzzy topology. We emphasize that fuzzy topological spaces naturally arise as spectrums of algebras of many-valued logics.

A Study of the Interrelation between Fuzzy Topological Systems and Logics

“…Definition 5.2.8 (Finite intersection property). [29] Let A be an L c n -algebra. A subset X of A has finite intersection property (f.i.p.)…”

“…By r we shall also denote the constant function with value r ∈n. The following definitions are standard ones restated from [29].…”

“…Definition 5.2.9 (Discreten-fuzzy topology). [29] For a set X,n X is called the discreten-fuzzy topology on X. (X,n X ) is called a discreten-fuzzy topological space.…”

“…On top of that duality between a special kind of fuzzy topological space whose value set is {0, 1 n − 1 , 2 n − 1 , 3 n − 1 , ......, n − 2 n − 1 , 1} and L c n is established. As a matter of fact another proof of duality provided in [29] comes into picture.…”

“…Exploration of relationship between many valued logics ( Lukasiewicz nvalued logic L c n ) and fuzzy topology has already been in the agenda [18,29]. We, in [18] (vide Chapter 5), have investigated the relationship between L c n and fuzzy topological systems but following [29], by L c n we have understood L c nalgebra. On top of that duality between a special kind of fuzzy topological space whose value set is {0, 1 n − 1 , 2 n − 1 , 3 n − 1 , ......, n − 2 n − 1 , 1} and L c n is established.…”