The Rough Intuitionistic Fuzzy Zweier Lacunary Ideal Convergence of Triple Sequence Spaces

Corresponding Author: Ayhan Esi Department of Mathematics, Adiyaman University, 02040, Adiyaman, Turkey Email: aesi23@hotmail.com Abstract: We introduced and studied the concept of I-convergence of triple sequences in metric spaces where I is an ideal. The concept of Iconvergence has a wide application in the field of number theory, trigonometric series, summability theory, probability theory, optimization and approximation theory. In this article, we introduce rough intuitionistic fuzzy Lacunary ideal convergent of triple sequence spaces via zwier operators. We discuss general topological properties.


Introduction
The idea of rough convergence was first introduced by Phu (2001;2002;2003) in finite dimensional normed spaces. He showed that the set r x LIM is bounded, closed and convex; and he introduced the notion of rough Cauchy sequence. He also investigated the relations between rough convergence and other convergence types and the dependence of r x LIM on the roughness of degree r. Aytar (2008a) studied of rough statistical convergence and defined the set of rough statistical limit points of a sequence and obtained two statistical convergence criteria associated with this set and prove that this set is closed and convex. Also, Aytar (2008b) studied that the r-limit set of the sequence is equal to intersection of these sets and that r-core of the sequence is equal to the union of these sets. Dündar and Cakan (2014) investigated of rough ideal convergence and defined the set of rough ideal limit points of a sequence The notion of I-convergence of a triple sequence spaces which is based on the structure of the ideal I of subsets of × × N N N , where N is the set of all natural numbers, is a natural generalization of the notion of convergence and statistical convergence.
Let K be a subset of the set of positive integers × × N N N and let us denote the set K ikℓ = {(m, n, k) ∈ K: m ≤ i, n ≤ j, k ≤ ℓ. Then the natural density of K is given by: where, |K ijℓ | denotes the number of elements in K ijℓ .
A triple sequence (real or complex) can be defined as a function x: ( ) × × → N N N R C , where N , R and C denote the set of natural numbers, real numbers and complex numbers respectively. The different types of notions of triple sequence was introduced and investigated at the initial by Sahiner et al. (2007;Tripathy, 2008) Esi (2014;Esi and Necdet Catalbas, 2014;Esi and Savas, 2015;Esi et al., 2016;2018), Dutta et al. (2013), Subramanian and Esi (2015), Debnath et al. (2015) and many others.
Let (x mnk ) be a triple sequence of real or complex numbers. Then the series A a = ℓ is an infinite four dimensional matrix of real or complex numbers mn k a ℓ , where k, ℓ, m, n∈ N . Then we say that A defines a matrix mapping from λ to µ and we do not it by writing A: λ→µ.
If for every sequence By (λ: µ), we denote the class of matrices A such that A: λ→µ. Thus a∈(λ,µ) if and only if (Ax)∈µ for every x∈λ.
The approach of constructing the new sequence spaces by means of the matrix domain of a particular limitation method.
The Z p -transformation of the sequence x = (x mnk ), (i.e.,) The Zweier sequence space 3 Z Γ as follows: In this study, we are working on to connect to studies which was mentioned above in order to introduce rough intuitionistic triple sequence spaces and study some algebraic and topological properties on these spaces.
A fuzzy number X is a fuzzy subset of the real 3 R ; which is normal fuzzy convex, upper semi-continuous and the X 0 is bounded where X 0 ;= cl{x∈ 3 R : X (x) > 0} and cl is the closure operator. These properties imply that for each α ∈(0, 1], the α-level set X α defined by: The supremum metric d on the set ( ) A subset E of ( ) 3 L R is said to be bounded above if there exists a fuzzy number µ, called an upper bound of E, such that X ≤ µ for every X∈E. µ is called the least upper bound of E if µ is an upper bound and µ ≤ µ′ for all upper bounds µ′.
A lower bound and the greatest lower bound are defined similarly. E is said to be bounded if it is both bounded above and below.
The notions of least upper bound and the greatest lower bound have been defined only for bounded sets of fuzzy numbers. If the set E⊂ ( ) 3 L R is bounded then its supremum and infimum exist.
The limit infimum and limit supremum of a triple sequence spaces (X mnk ) is defined by: Where: Now, given two fuzzy numbers X, Y∈ ( ) To any real number 3 a ∈ R , we can assign a fuzzy number A set E of fuzzy numbers is called convex if λµ 1 + (1-λ)µ 2 ∈E for all λ∈ [0, 1] and µ 1 , µ 2 ∈E.

Definitions and Preliminaries
In this section, we present some preliminary definitions and results related to intuitionistic fuzzy metric triple sequence spaces and that will be used throughout the article.

Definition 2.1
The five-tuple (X, µ, v, * , ◊) is said to be an Intuitionistic Fuzzy Metric Space (for short, IFMS) if X is a vector space, * is a continuous t-norm, ◊ is a continuous t-conorm and µ, v are fuzzy sets on X ×(0,∞) satisfying the following conditions for every x; y ∈ x and s, t > 0: In this case (µ, v) is called intuitionistic fuzzy metric.

Definition 2.2
Let (X, µ, v, * , ◊) be an rough intuitionistic fuzzy metric space and β be a non negative real number. A triple sequence x = (x m nk) in X is said to be convergent to L∈X with respect to the intuitionistic fuzzy metric (µ, v) if, for every ϵ ∈(0, 1) and

Definition 2.3
Let (X, µ, v, * , ◊) be an rough intuitionistic fuzzy metric space and β be a non negative real number. A triple sequence x = (x mnk ) in X is said to be a Cauchy sequence with respect to the rough intuitionistic fuzzy metric (µ, v) if, for every ϵ ∈(0, 1) and t > 0, there exists m 0 n 0 k 0 ∈ N such that µ(x mnk -x uvw , t) > 1 -(β + ϵ) and v(x mnk −x uvw , t) < β + ϵ for all m, n, k, u, v, w ≥ m 0 n 0 k 0 .

Definition 2.4
Let (X, µ, v, * , ◊) be an rough intuitionistic fuzzy metric space. For t>0, the open ball B(x, r, t) with center x∈ X and radius r ∈ (0, 1) is defined as: If X be a non-empty set. Then a family I⊂ P(X) of subsets of X is called an ideal in X if and only if: where, P(X) is the power set of X.

Definition 2.6
If X be a non-empty set. Then a non-empty family of sets F⊂ P(X) is called a filter on X if and only if:

Remark 2.1
For each ideal I there is a filter F (I) which corresponding to I (filter associate with ideal I), that is:

Definition 2.7
The triple sequence θ i,ℓ,j = {(m i , n ℓ , k j )} is called triple lacunary if there exist three increasing sequences of integers such that: Definition 2.8 Let I⊂ ( ) P N and let (X, µ, v, * , ◊) be an rough intuitionistic fuzzy metric space and β be a non negative real number. A triple sequence x = (x mnk ) in X is said to be Lacunary I-convergent to L∈X with respect to the rough intuitionistic fuzzy metric (µ, v) if, for every ϵ∈(0, 1) and t > 0, the set: L is called the Lacunary I-limit of the triple sequence x = (x mnk ) and we write ( ) ,v I µ θ -lim x = L.

Definition 2.9
Let (X, µ, v, * , ◊) be an rough intuitionistic fuzzy metric space and β be a non negative real number. A triple sequence x = (x mnk ) in X is said to be Lacunary I-Cauchy with respect to the rough intuitionistic fuzzy metric (µ, v) if, for every ϵ∈(0, 1) and t > 0, there exists m ∈ N satisfying: The Zweier operator and I θ -convergence of triple sequences in an rough intuitionistic fuzzy metric spaces and β be a non negative real number, we introduce the following new Zweier triple sequence spaces and examine some algebraic and topological properties on these spaces. Throughout the article, for the sake of convenience now we will denote by Z p (x mnk ) = x′, Z p (y mnk ) = y′, Z p (z mnk ) = z′ where x = (x mnk ), y = (y mnk ), z = (z mnk ) are in ω: Also, we define an open ball with center x′ and radius r with respect to t as follows: Define the set A 3 = A 1 ∩A 2 , so that A 3 ∈I. it follows that 3 c A is a non-empty set in F(I). We shall show that for each (x′ mnk ), (y′ mnk )∈ ( ) ∈ . In this case:
for each x∈A, ∃t > 0 and