LATTICE VALUED NEUTROSOPHIC SETS

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this paper, the authors introduce lattice valued neutrosophic sets and study their properties. A decomposition theorem for lattice valued neutrosophic sets is obtained. Lattice valued neutrosophic mappings are defined and verified that its properties are consistent with their crisp counterparts. Finally, a topology of lattice valued neutrosophic sets is introduced.


INTRODUCTION
Zadeh's [24](1965) fuzzy set theory and fuzzy logic brought wide applications in the domain of uncertainities. Fuzzy sets successfully handled cases where an element partially belongs to a set. But fuzzy sets could not handle those cases where uncertainities arose due to incomplete data. Such a scenario inspired the introduction of intuitionistic fuzzy sets by Atanassov [1].
Though intuitionistic fuzzy set theory was efficient in modeling incomplete information, it could not handle indeterminate and inconsistent data.
In 1998, Smarandache [19] proposed the concept of neutrosophic sets, which generalizes fuzzy sets, interval valued fuzzy sets, intuitionistic fuzzy sets and interval valued intuitionistic fuzzy sets. In addition to the truth-membership and the falsity-membership, an independent indeterminancy membership function defines a neutrosophic set. Several extensions of neutrosophic sets such as interval valued neutrosophic sets [21], bipolar neutrosophic sets [4], neutrosophic soft sets [15] etc, were studied by researchers to deal with a variety of problems.
Neutrosophic set theory proved to have wide applications in decision making problems [16] and medical image processing [12]. Single valued neutrosophic sets were introduced by Wang et al. [20]. It became a hot area of research due to its applicability to practical problems [9,2]. Single valued neutrosophic relations were studied by Kim et al [10]. In 2005, Smarandache defined various notions of neutrosophic topologies [18]. Kim, J. et al [11] studied ordinary single valued neutrosophic topologies. M. EL-Gayyar [5] introduced smooth neutrosophic topological spaces.
Chang's [3] introduction of fuzzy set theory into topology initiated extensive research in fuzzy set theory. Goguen [7] replaced the unit interval in a fuzzy set by a lattice to define L-fuzzy sets and subsequently introduced L-fuzzy topology [8](known as the Chang-Goguen L-fuzzy topology). Later, several authors looked into the interaction between lattice theory and topology in different directions.
In this paper, we introduce lattice valued neutrosophic sets and evaluate its basic properties.
We show that a lattice valued neutrosophic set can be decomposed into level subsets. Lattice valued neutrosophic mappings and inverse lattice valued neutrosophic mappings are defined to connect different lattice valued neutrosophic sets. In the final section, a topology of lattice valued neutrosophic sets is introduced.

PRELIMINARIES
In this section, a brief overview of neutrosophic set theory is provided. Essential concepts and results from lattice theory are also discussed.

Neutrosophic Set Theory. Since the introduction of neutrosophic sets by
Smarandache [19], various authors introduced different types of operations on neutrosophic sets and studied their properties. LATTICE VALUED NEUTROSOPHIC SETS 4697 In this paper, we choose only the most naturally defined types of operations, which agrees with human intuition.
Definition 2.1. [19](Neutrosophic set) Let X be a set, with a generic element in X denoted by x.
A neutrosophic set A in X is characterized by three membership functions: a truth membership function T A , an indeterminancy membership function I A and a falsity membership function F A , where ∀x ∈ X, T A (x), I A (x) and F A (x) are real standard or non-standard subsets of ]0 − , 1 + [.
There is no restriction on the sum of T A (x), I A (x) and F A (x).
Definition 2.5. [17] Let A and B be two SVN sets in X.
(1) The union of A and B is a SVN set C, denoted C = A ∪ B, where ∀x ∈ X Definition 2.6. [13](Lattice) Let L be a poset. L is called a lattice if any two of its elements a and b have a greatest lower bound ("meet") denoted by a ∧ b and a least upper bound ("join") denoted by a ∨ b. A lattice L is said to be complete when each of its subsets has an l.u.b and g.l.b in L. In particular, the smallest element 0 L and the greatest element 1 L will exist in L as the join of the empty set and the meet of the empty set respectively. Therefore, all lattices we consider in this paper are assumed to contain atleast 0 L and 1 L .
The following notions are from [6]. An element l ∈ L \ {0 L } is called a co-prime element if, for any finite subset K ⊂ L satisfying l ≤ K, ∃ k ∈ K such that l ≤ k. The set of all co-prime elements of L will be denoted by c(L). An element l ∈ L \ {1 L } is called a prime element if it's a co-prime element of L op . The set of all prime elements of L will be denoted by p(L).
There exists a stronger form of inequality in a lattice, known as 'way below relation'. A point Theorem 2.11. [6] If L is completely distributive, then c(L) is a join-generating set of L and p(L) is a meet-generating set of L. i.e., every element in L is the supremum of all the co-primes way below it and the dual statement also holds.

LATTICE VALUED NEUTROSOPHIC SETS
In this section we introduce the concept of Lattice Valued Neutrosophic (LVN) set. Basic operations on LVN sets are defined and their respective properties are investigated. The introduced operations behave analogous to their crisp counterparts. All lattices considered in the following sections are assumed to be complete and infinitely distributive.
LVNS(X) will denote the set of all lattice valued neutrosophic sets on X. Let A ∈ LV NS(X), Two LVN sets are said to be equal if and only if A ⊆ B and A ⊇ B.
The assumption of an order reversing involution enables us, first of all, to give a reasonable definition for closedness and some related notions.
Definition 3.5. Let A be an LVN set in X. The complement of A is denoted by A c , where ∀x ∈ X, Definition 3.6. Let A and B be two LVN sets in X.
(1) The union of A and B is a LVN set C, denoted C = A ∪ B, where ∀x ∈ X, (2) The inersection of A and B is a LVN set D, denoted D = A ∩ B, where ∀x ∈ X, The lattice L being infinitely distributive, arbitrary union and arbitrary intersection can be defined in the obvious way.
Theorem 3.7. The following properties hold when A, B,C ∈ LV NS(X).
Theorem 3.9. The neutrosophic empty set and full set satisfies the following equalities:  (1) L is infinitely distributive iff LVNS(X) is infinitely distributive.
Definition 3.15. Suppose we are given a crisp set A, we can convert A into a LVN set N(A) as follows: In particular, for a given (a,b,c)-level A a,b,c of a LVN set A, N(A a,b,c ) is as follows: Thus, the LVN set (a, b, c)N(A a,b,c ) will have the following truth, indeterminancy and falsity values: Let x ∈ X and (T A (x), I A (x), F A (x)) = (α 1 , α 2 , α 3 ).
We can do better if we assume that L is completely distributive, in that case, since c(L) is a join generating set and p(L) is a meet generating set, the following statement holds: Theorem 3.17. Let L be a completely distributive lattice and for a ∈ c(L) and b, c ∈ p(L) takẽ

LATTICE VALUED NEUTROSOPHIC MAPPINGS
In this section, we define the concept of lattice valued neutrosophic mappings(LVN mappings) and inverse lattice valued neutrosophic mappings. Some of their properties are also investigated. (1) Define f → as a LVN mapping from LVNS(X) to LVNS(Y) induced by f as (2) Define f ← as a inverse LVN mapping from LVNS(Y) to LVNS(X) induced by f as It's easy to observe that the notion of LVN mappings is a generalization of SVNR mappings introduced by Yang [22] Theorem 4.2. Let A 1 , A 2 ∈ LV NS(X), B 1 , B 2 ∈ LV NS(Y ) and f be a function from X to Y. The following properties are true: Theorem 4.3. Let A ∈ LV NS(X), B ∈ LV NS(Y ) and f be a function from X to Y. The following hold: (1) f → ( f ← (B)) ⊆ B, equality holds if f is a surjection.
(2) f ← ( f → (A)) ⊇ A, equality holds if f is an injection. Proof: . Equality holds if f is injective.

Proof:
For x ∈ X and y ∈ Y , The following results hold: (

the identity function on LVNS(X).
Theorem 4.6. Let f : X → Y and g : Y → Z be ordinary functions. Then (2) f ← • g ← = (g f ) ← Definition 4.7. An LVN Point on X is an LVN set x (a,b,c) ∈ LV NS(X) defined by The tuple (a, b, c) is called the height of the LVN point x (a,b,c) .
One can easily observe that any given LVN set can be written as the union of LVN points contained in it. Proof: We will first show that there exists a function f : Since F preserves LVN points, ∀x ∈ X and ∀(a, b, c) Thus we have obtained an ordinary function f : X → Y such that The uniqueness of f is clear.

LATTICE VALUED NEUTROSOPHIC TOPOLOGY
In this section, we define a new topological structure which is a generalization of the single valued neutrosophic topology. The lattice valued neutrosophic counterpart of the interior operator and closure operator are introduced and their basic properties are listed.   Proof: (1) First we prove the necessity part. Let τ be the unique topology generated by A . Then, the necessity part follows since A is a subset of a topology. Conversely, take τ = {∪C : The empty LVN set belongs to τ as the union of empty class from A . By definition, τ is closed under arbitrary union. (2) Follows from (1).
Definition 5.4. Let (X, τ) be a LVN topological space and A ∈ LV NS(X).
(1) The LVN interior of A is defined as LNint(A) = ∪{A i : A i ∈ τ, A i ⊆ A}

CONCLUSION
The study have introduced a generalization of single valued neutrosophic sets and a corresponding topological structure. Various properties of the introduced concepts were found to be in consistent with their conventional counterparts.
Investigating the neighbourhood structures of LVN topological spaces forms part of our future study. The relations of the generalized structures with their special cases will be examined in the light of category theory. Having defined the category of lattice valued neutrosophic sets, its relation with various subcategories can be investigated and a similar investigation for the category of LVN topological spaces can also be undertaken.