eu L-CLOSURE OPERATORS AND L-FUZZY PRE-PROXIMITIES

Hájek [7] introduced a complete residuated lattice which is an algebraic structure for many valued logic. It is an important mathematical tool for algebraic structure. By using the concepts of lower and upper approximation operators, information systems and decision rules are investigated in complete residuated lattices [3,4,6,15,19]. Höhle [8] introduced L-fuzzy topologies with algebraic structure L(cqm, quantales, MV -algebra). Katsaras [9-11] introduced the L-fuzzy proximity spaces in complete distributive lattices. Kim [13] extended the the L-fuzzy proximity spaces in strictly two-sided commutative quantales and investigated their topological properties.


Introduction
Hájek [7] introduced a complete residuated lattice which is an algebraic structure for many valued logic.It is an important mathematical tool for algebraic structure.By using the concepts of lower and upper approximation operators, information systems and decision rules are investigated in complete residuated lattices [3,4,6,15,19].Höhle [8] introduced L-fuzzy topologies with algebraic structure L(cqm, quantales, M V -algebra).
Katsaras [9][10][11] introduced the L-fuzzy proximity spaces in complete distributive lattices.Kim [13] extended the the L-fuzzy proximity spaces in strictly two-sided commutative quantales and investigated their topological properties.
In this paper, we introduce the notions of L-fuzzy pre-proximities and Lclosure operators in complete residuated lattices.We obtain the L-closure operators induced by L-fuzzy pre-proximities in Theorem 8 and the L-fuzzy preproximities induced by L-closure operators in Theorem 9.Moreover, we study the relations between the L-fuzzy pre-proximities and L-closure operators.We give their examples.
An L-fuzzy pre-proximity is called an Lemma 6. [6,14] For a given set X, define a binary mapping S : Then, for each f, g, h, k ∈ L X , and α ∈ L, the following properties hold.
(1) S is an L-partial order on L X . ( and the equalities hold if φ is bijective.
3. L-Fuzzy Pre-Proximities and L-Closure Operators Lemma 7. Let C : L X → L X a map.The following statement are equivalent.
(1) For all f, g ∈ L X , S(f, g) ≤ S(C(f ), C(g)). ( From the following theorem, we obtain the L-closure operator induced by an L-fuzzy pre-proximity.Theorem 8. Let (X, δ) be an L-fuzzy pre-proximity space.We define a mappings Then C δ is a stratified L-closure operator on X.
From Lemma 6, we obtain (C3) and by Lemma 7, C δ is stratified from: (by Lemma 2 ( 16)) Hence C δ is a stratified L-closure operator on X.
From the following theorem, we obtain the L-fuzzy pre-proximity induced by an L-closure operator.Theorem 9. Let (X, C) be an L-closure space.Define a mapping δ C : Then we have the following properties.