Abstract
We introduce the notion of (dual) residuated frames as a viewpoint of relational semantics for a fuzzy logic. We investigate the relations between (dual) residuated frames and (dual) residuated connections as a topological viewpoint of fuzzy rough sets in a complete residuated lattice. As a result, we show that the Alexandrov topology induced by fuzzy posets is a fuzzy complete lattice with residuated connections. From this result, we obtain fuzzy rough sets on the Alexandrov topology. Moreover, as a generalization of the Dedekind–MacNeille completion, we introduce R-R (resp. D R - D R ) embedding maps and R-R (resp. D R - D R ) frame embedding maps.
Highlights
Blyth and Janovitz [1] introduced the residuated connection as a pair ( f, g) of maps from a partially ordered set ( X, ≤ X ) to a partially ordered set (Y, ≤Y ) such that for all x ∈ X, y ∈ Y, f ( x ) ≤Y y if and only if x ≤ X g(y)
As a duality between algebras and logical relational systems, we introduce the notion of residuated connections and residuated frames in fuzzy logics
Where τeX and τeY are Alexandrov L-topologies induced by fuzzy posets ( X, eX ) and (Y, eY ) in Theorem 1
Summary
Blyth and Janovitz [1] introduced the residuated connection as a pair ( f , g) of maps from a partially ordered set ( X, ≤ X ) to a partially ordered set (Y, ≤Y ) such that for all x ∈ X, y ∈ Y, f ( x ) ≤Y y if and only if x ≤ X g(y). Rough sets form residuated connections in the following sense: let R be an equivalence relation on X. Fuzzy rough sets form residuated connections in the following sense: for all A, B ⊂ X, e LY ( F ( A ) , B ) =. As a duality between algebras and logical relational systems, we introduce the notion of residuated connections and residuated frames in fuzzy logics. Y ∈Y where τeX and τeY are Alexandrov L-topologies induced by fuzzy posets ( X, eX ) and (Y, eY ) in Theorem 1. Using this result, one can show that the pair ( F ( A), G ( A)) is an fuzzy rough set for.
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