Notes on “Some Properties of L -fuzzy Approximation Spaces on Bounded Integral Residuated Lattices”

: In this note

A residuated lattice with a constant 0 is called an FL-algebra. If 1 x ≤ for all x L ∈ , then L is called integral residuated lattice. An FL-algebra L, which satisfies the condition 0 1 x ≤ ≤ for all x L ∈ , is called an w FL -algebra or a bounded integral residuated lattice (see [1]).
We adopt the usual convention of representing the monoid operation by juxtaposition, writing ab for a b ⋅ .
Let L be a bounded integral residuated lattice. Define two negations on L, L ¬ and R ¬ : A bounded residuated lattice L is called an involutive residuated lattice (see [3] In the sequel, unless otherwise stated, L always represents any given complete involutive residuated lattice with maximal element 1 and minimal element 0. Definition 1.1 (see Liu and Luo [5]). Let X L τ ⊆ and J be an index set. If τ satisfies the following three conditions: then τ is called an L-fuzzy topology on X and ( , ) int , L cl and R cl are, respectively, called the interior, left closure and right closure operators.
For the sake of convenience, we denote int( ) Zhang et al. [14,15] studied some properties of rough sets and rough approximation operators, Ouyang et al. [6,7] investigated some generalized models of fuzzy rough sets, Liu and Lin [4] considered the intuitionistic fuzzy rough set model, Wu et al. [13] discussed the axiomatic characterizations of fuzzy rough approximation operators, Radzikowska and Kerre [9] studied L-fuzzy rough sets and lower (upper) L-fuzzy approximation based on commutative residuated lattices. Recently, 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. Definition 1.3 (Wang et al. [12]). Let R be an L-relation on X . A pair ( , ) X R is called an L-fuzzy approximation space. Define the following four mappings , L-fuzzy rough approximation operators, L-fuzzy approximation space and left (right) L-fuzzy rough sets are, respectively, called fuzzy rough approximation operators, fuzzy approximation space and left (right) fuzzy rough sets.

The L-fuzzy Topologies Generated by a Reflexive and Transitive L-relation
In this section, we supplement some properties of the L-fuzzy topologies generated by a reflexive and transitive L-relation.
If R is a reflexive and transitive L-relation on X , then it follows from Theorem 6.1 in are all L-fuzzy topologies on X and  Proof. When L is an involutive residuated lattice, If R is a reflexive and transitive L-relation on X , then it follows from Theorem 4.1(5) and Remark 5.
∈ and X L µ ∈ , then it follows from Theorems 3.1(3) and 4.1(5) in The theorem is proved.
Recently, Qin et al. [2,8] studied the topogical properties of fuzzy rough sets. The following left and right (TC) axioms are generalizations of (TC) axiom in [8].
The theorem is proved.

The L-relations Induced by an L -fuzzy Topology
In this section, we supplement some properties of the L-relations induced by an L-fuzzy topology.
It is easy to see that is also a reflexive and transitive L-relations on X . Therefore, The theorem is proved. This result shows that the reflexive and transitive L-relations Thus, by Definition 1.3 and Theorem 4.1(3) in [12], we see that Let τ be an L-fuzzy topology on X and J index set. Then the following properties hold.
(1) If τ satisfies (TC) L axiom and the left closure operator w.r.t. τ satisfies the following two conditions: , , (2) If τ satisfies (TC) R axiom and the right closure operator w.r.t. τ satisfies the following two conditions: for any x X ∈ . Thus, for any i.e., L L R τ ↑ is just the left closure operator w.r.t. τ . By Theorems 3.1(2) and 4.1(5) in [12], we see that i.e., L R R τ ↓ is just the interior closure operator w.r.t. τ . Therefore, The theorem is proved. This result shows that the L-topologies generated by two reflexive and transitive L-relations L R τ and R R τ , which are induced by an L-topology τ , on X are all the original L-topology τ when τ satisfies some conditions. Moreover

Conclusions and Future Work
In this note, we continue the works in [12]. For a complete involutive residuated lattice, we have supplemented some properties of the L-fuzzy topologies generated by a reflexive and transitive L-relation; showed that the L-fuzzy topologies generated by a reflexive and transitive L-relation satisfy (TC) L or (TC) R axioms; and given out some conditions such that the L-fuzzy topologies generated by two L-relations, which are induced by an L-fuzzy topology, are all the original L-fuzzy topology.
In a forthcoming paper, we will discuss the relationships between the L-fuzzy topological spaces and the L-fuzzy rough approximation spaces on a complete involutive residuated lattice.