SOME GENERALIZATIONS OF L-CLOSED SET

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this paper we study and characterize L-subsets like p-locally L-closed sets, λ -locally L-closed sets, Λλ L-closed sets, gL-closed sets, λgL-closed. Further we define and study p-LC-L-continuity, λ -LC-L-continuity and Λλ -L-continuity and we obtain decompositions of L-continuity.


INTRODUCTION
The concept of fuzzy set was introduced by Zadeh [13] in his classical paper. Fuzzy topology was introduced by Chang [3] in 1965. Subsequently, many researchers have worked on various basic concepts from general topology using fuzzy sets and developed the theory of fuzzy topological spaces. There are many applications of fuzzy sets in different fields such as information theory [10] and control problems [11]. Levine [5] initiated the study of generalized closed sets in topological space in order to extend many of the important properties of closed sets to a larger family. Locally closed set is one of the generalization of closed set. The first step of locally closedness was done by Bourbaki [2].
Ganster and Reily [4] used locally closed set to define LC-continuity and LC-irresoluteness.
Several mathematicians generalized this notion by replacing open sets with nearly open sets and/or by replacing closed sets with nearly closed sets. In 1997, G. Balasubramanian and P.
Sundaram [1] defined generalized fuzzy closed sets in fuzzy topology. Later many mathematicians extented different generalization of closed sets to fuzzy topology.

PRELIMINARIES
In this section, we include certain definitions and known results needed for the subsequent development of the study. Throughout this paper, X be a non empty ordinary set and (L, ) be a Hutton algebra, that is, a complete, completely distributive lattice L equipped with an orderreversing involution. SOME GENERALIZATIONS OF L-CLOSED SET 6531 Definition 2.1. [6] Let L be a complete lattice, C ⊆ L then C is a join generating set if ∀a ∈ L, ∃C a ⊂ C such that ∨C a = a.  An L-point on X with support x and value λ is an L-subset x λ ∈ L X defined as Let X be a non-empty ordinary set and (L, ) be a Hutton algebra and δ ⊆ L X .
Then δ is called an L-topology on X and (X, δ ) is called an L-toplogical space, if δ satisfies the following three conditions:

p-LOCALLY L-CLOSED
Remark 3.1. The join and meet of two p-locally L-closed set is not generally p-locally L-closed.
Clearly, δ = {0, 1, a m , a n , a 1 ,U,V } is an L-topology on X. It is easy to see that a m , a n ,U,V are p-locally L-closed but U ∧V = a 1 and a m ∨ a n = a 1 are not p-locally L-closed.
Proposition 3.1. Let (X, δ ) be an L-topological space then the following hold: 1) Every L-closed set is p-locally L-closed set.
2) Every pL-open set is p-locally L-closed.
3) Every p-locally L-closed is locally L-closed.
2) Follows from the definition.   (1) A is p-locally L-closed,     Proof. Every L-closed set is clearly both p-locally L-closed and gL-closed. For converse part suppose that A be gL-closed and p-locally L-closed. Since A is p-locally L-closed, by lemma     Proof. Suppose that A is λ -locally L-closed. Then we can find a λ L-open set U and an L- Proposition 4.3. Let (X, δ ) be an L-topological space then the following hold: 1) Every L-closed set is λ -locally L-closed set.
3) Every p-locally L-closed set is λ -locally L-closed.   (1) A is λ -locally L-closed,     Let (X, δ ) be an L-topological space and A, B ∈ L X . Then the following hold: Let (X, δ ) be an L-topological space and A ∈ L X then the following hold:  (1) Every λ -locally L-closed set is Λ λ L-closed, (2) Every p-locally L-closed set is Λ λ L-closed.
Proof. 1) Let A be a λ -locally L-closed set. Then there exist a λ L-open set U and L-closed F such that A = U ∧ F. By Lemma 4.2, U is a Λ λ L-set and therefore A is Λ λ L-closed.

DECOMPOSITIONS OF L-CONTINUITY
In this section we obtain some decompositions of L-continuity.   2) Λ λ L-continuous if f −1 (F) is Λ λ L-closed in (X, δ ) for any L-closed set F in (Y, µ).
3) λ gL-continuous if f −1 (F) is λ gL-closed in (X, δ ) for any L-closed set F in (Y, µ). Then F is L-closed but f −1 (F) is not λ -locally L-closed. Therefore f is not λ -LC-L-continuous but it is easy to see that f is Λ λ L-continuous.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.