Abstract
In this note, we continue the works in the paper [Some properties of <i>L</i>-fuzzy approximation spaces on bounded integral residuated lattices", Information Sciences, 278, 110-126, 2014]. For a complete involutive residuated lattice, we show that the <i>L</i>-fuzzy topologies generated by a reflexive and transitive <i>L</i>-relation satisfy (TC) <sub><i>L</i></sub> or (TC) <sub><i>R</i></sub> axioms and the <i>L</i>-relations induced by two <i>L</i>-fuzzy topologies, which are generated by a reflexive and transitive <i>L</i>-relation, are all the original <i>L</i>-relation; and give out some conditions such that the <i>L</i>-fuzzy topologies generated by two <i>L</i>-relations, which are induced by an <i>L</i>-fuzzy topology, are all the original <i>L</i>-fuzzy topology.
Highlights
IntroductionAn FL-algebra L, which satisfies the condition 0 ≤ x ≤ 1 for all x ∈ L , is called an FLw -algebra or a bounded integral residuated lattice (see [1])
A residuated lattice is an algebra L = (L, ∧, ∨, ⋅, →, ←, 0, 1) of type (2, 2, 2, 2, 2, 0, 0)satisfyingthe following conditions: (L1) (L, ∧, ∨) is a lattice,(L2) (L, ⋅, 1) is a monoid, i.e., is associative and x ⋅1 = 1⋅ x = x for any x ∈ L,(L3) x ⋅ y ≤ z if and only if x ≤ y → z if and only if y ≤ x ← z for any x, y, z ∈ L .A residuated lattice with a constant 0 is called an FL-algebra
If R is a reflexive and transitive L-relation on X, it follows from Theorem 6.1 in [12] that τ1 = {ξ | R ↓L (ξ ) = ξ, ξ ∈ LX }, τ 2 = {ξ | R ↓R (ξ ) = ξ, ξ ∈ LX }
Summary
An FL-algebra L, which satisfies the condition 0 ≤ x ≤ 1 for all x ∈ L , is called an FLw -algebra or a bounded integral residuated lattice (see [1]). Let L be a bounded integral residuated lattice. 11 Yuan Wang et al.: Notes on “Some properties of L-fuzzy Approximation Spaces on Bounded Integral Residuated Lattices”. Wang et al [12] discussed the notion of left (right) lower and left (right) upper L-fuzzy rough approximation based on complete bounded integral residuated lattices. Define the following four mappings R ↓L , R ↑L , R ↓R , R ↑R : LX → LX , called a left lower, left upper, right lower, and right upper L-fuzzy rough approximation operators, respectively, as follows: for every μ ∈ LX and x∈X ,.
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