Steinhaus [1] and Fast [2] first proposed the concept of statistical convergence, which was subsequently examined and developed by several authors. Suja and Srinivasan [3] presented statistical convergence and Cauchy sequences over non-Archimedean fields. The analysis of non-Archimedean fields is known as non-Archimedean analysis. The concept of n-isometries in non-Archimedean n-normed spaces was first studied by Chu and Ku [4]. In 1965, Zadeh [5] developed the fuzzy set theory in a fuzzy-normed linear space, statistically convergent sequences, statistically Cauchy sequences, statistical limit points, and statistical cluster points of a sequence were introduced by Sencimen and Pehlivan [6].

Mohiuddine et al. [7] studied the statistical convergence of double sequences in fuzzy normed spaces. The n-normed linear space to a fuzzy n-normed linear space by Narayanan and Vijayabalaji [8]. Saadati and Park [9] introduced the concept of an intuitionistic fuzzy normed space. Karakus et al. [10] recently studied the concept of statistical convergence of single sequences in an intuitionistic fuzzy normed space, and Mursaleen and Mohiuddine [11] extended it to double sequences.

Mohiuddine and Lohani [12] studied generalized statistical convergence in an intuitionistic fuzzy normed space, which provides a better tool for analyzing a broader class of sequences. Karakaya et al. [13] investigated the types of convergence of function sequences in intuitionistic fuzzy-normed spaces. Melliani et al. [14] defined the intuitionistic fuzzy deferred statistical convergence in an intuitionistic fuzzy normed space by considering the deferred density.

Generalized statistical convergence in a non-Archimedean L-fuzzy normed space was investigated by Eghbati and Ganji [15]. The idea of statistically convergent and statistically Cauchy sequences in intuitionistic fuzzy 2-normed spaces was first explored by Alotaibi [16] and later expanded by Savas [17] to ideal lambda-statistical convergence in these spaces.

Vijayabalaji et al. [18] and Sen and Debnath [19] introduced the concepts of statistical convergence and statistical Cauchy sequences in intuitionistic fuzzy n-normed linear spaces (IFnNLS). Konwar and Debnath [20] extended the concept of IFnNLS to define an intuitionistic fuzzy n-normed algebra (IFnNA).

The objective here is to present and explore the concepts of statistical convergence and statistical Cauchy sequence in an intuitionistic fuzzy n-normed space and to extract some useful conclusions about them. Additionally, we present statistical completeness as a paradigm for investigating the fullness of intuitionistic fuzzy n-normed spaces.

In this study, we consider a sequence in the non-archimedean field K. Throughout this paper, K denotes a nontrivially valued complete non-archimedean field. We begin by recalling several notations and definitions used throughout this paper.

Definition 2.1

Let X be a vector space with the dimension d ≥ n over a valued field K with a non-Archimedean valuation |.|.

A function ||., ., ., ..., .|| : X × X × . . .X → [0,∞) is said to be a non-Archimedean n-norm if

• (i) ||x₁, x₂, …, x_n|| = 0 ⇔ x₁, x₂, …, x_n are linearly dependent,

• (ii) ||x₁, x₂, …,x_n|| = ||x_r1 , x_r2 , …, x_rn|| for every permutation (r₁, r₂, …, r_n) of (1, 2, …n),

• (iii) ||ax₁, x₂, …, x_n|| = |a| ||x₁, x₂, …, x_n||,

• (iv) ||x + y, x₂, x₃, …, x_n|| ≤ max{||x, x₂, …, x_n||, ||y, x₂, …x_n||} for all α ∈ K and x, y, x₂, …, x_n ∈ X. Then (X, ||., ., ., ..., .||) is a non-Archimedean n-normed space.

Example 2.2

Let X = Rⁿ be equipped with the Euclidean n-norm

where x_j = (x_j1, ..., x_jn) ∈ Rⁿ for each j = 1, 2, ..., n.

Definition 2.3

A binary operations * : [0, 1]×[0, 1] → [0, 1] is said to be a continuous t-norm, if

• (a) * is associative and commutative.

• (b) * is continuous.

• (c) a * 1 = a for all a ∈ [0, 1].

• (d) a * b ≤ c * d, whenever a ≤ c and b ≤ d for each a, b, c, d ∈ [0, 1].

Definition 2.4

A binary operation ⋄ : [0, 1] × [0, 1] is considered a continuous t-conorm if

• (a’) ⋄ is associative and commutative.

• (b’) ⋄ is continuous.

• (c’) a ⋄ 0 = a for all a ∈ [0, 1].

• (d’) a ⋄ b ≤ c ⋄ d whenever a ≤ c and b ≤ d for each a, b, c, d ∈ [0, 1].

Definition 2.5

The intuitionistic fuzzy n-normed spaces over K is defined as follows. The five-tuple (X, η, φ, *, ⋄) is said to be an NA-IFnN space if V is a vector space over a field , ⋄ is a continuous t-conorm, * is a continuous t-norm, and η, φ are functions of X × ℝ to [0, 1] satisfying the following conditions. For every (x₁, x₂, ....x_n) ∈ Xⁿ and u, v ∈ .

• (i) η(x₁, x₂, ....x_n, u) + φ(x₁, x₂, ....x_n, u) ≤ 1.

• (ii) η(x₁, x₂, ....x_n, u) > 0.

• (iii) η(x₁, x₂, ....x_n, u) = 1 ⇔ x₁, x₂, ..., x_n are linearly dependent.

• (iv) η(x₁, x₂, ....x_n, u) is invariant under any permutation of x₁, x₂, ..., x_n.

• (v) η(x1,x2,....c.xn,u)=η(x1,x2,....xn,u∣c∣) for c ≠ 0, c ∈ K.

• (vi) η(x1,x2,…,xn,u)*η(x1,x2,…,xn′,v)≤μ(x1,x2,…,xn+xn′,max{u,v}).

• (vii) η(x₁, x₂, ..., x_n, ·) : (0,∞) → [0, 1] is continuous.

• (viii) limu→∞η(x1,x2,…,xn,u)=1 and limu→0η(x1,x2,…,xn,u)=0.

• (ix) φ(x₁, x₂, ..., x_n, u) < 1.

• (x) φ(x₁, x₂, ..., x_n, u) = 0 ⇔ x₁, x₂, ..., c·x_n are linearly dependent.

• (xi) φ(x₁, x₂, ..., x_n, u) is invariant under any permutation of x₁, x₂, ..., x_n.

• (xii) φ(x1,x2,....c.xn,u)=φ(x1,x2,....c.xn,u∣c∣) for each c ≠ 0, c ∈ K

• (xiii) φ(x1,x2,…,xn,u)⋄φ(x1,x2,…,xn′,v)≥φ(x1,x2,…,xn+xn′,max{u,v}).

• (xiv) φ(x₁, x₂, ..., x_n, ·) : (0,∞) → [0, 1] is continuous.

• (xv) limu→∞φ(x1,x2,…,xn,u)=0 and limu→0φ(x1,x2,…,xn,u)=1.

Here, (η, φ) is called a non-Archimedean intuitionistic fuzzy n-norm.

Example 2.5

Let (X, ||.||) be an n-normed space. Also let a * b = ab and a ⋄ b = min{a + b, 1} for all a, b ∈ [0, 1], η(x1,x2,…,xn)=tt+‖x1,x2,…,xn‖ and φ(x1,x2,…,xn,u)=‖x1,x2,…,xn‖t+‖x1,x2,…,xn‖. Thus, (X, η, φ, *, ⋄) is the IFnNS.

Definition 2.6

A sequence x = {x_k} in a NA-IFnNS (X, η, φ, *, ⋄) is considered convergent to ℓ ∈ X with respect to NA-IFnNS (η, φ)ⁿ, if for every ε > 0, u > 0 and y₂, y₃, ..., y_n ∈ X, there exist n₀ ∈ ℕ such that η(x_k − ℓ, y₂, y₃, ..., y_n, u) > 1 − ε and φ(x_k − ℓ, y₂, y₃, ..., y_n, u) < ε for all k ≥ k₀. This is denoted by (η, φ)ⁿ − limx = ℓ.

Definition 2.7

A sequence x = {x_k} is considered statistically convergent to the number ℓ if, for every ε > 0, limn1n∣k∈ℕ:∣xk-l∣≥ε∣=0. In this case, we write Stat–limx = ℓ.

Example 2.8

Consider the sequence x = {x_k} defined by

By choosing the non-Archimedean valuation to be 2-adic, the terms of the sequence are (0, 0, 0, 1, 0, 0, 0, 0, 1/8, 0, 0, ...). This sequence statistically converged to 0.

Definition 3.1

Let (X, η, φ, *, ⋄) be a NA-IFnNS. A sequence x = {x_k} in X is said to be statistically convergent to ℓ ∈ X with respect to the intuitionistic fuzzy n-norm (η, φ)ⁿ if, for every ε > 0, u > 0 and y₂, y₃, ..., y_n ∈ X,

In this case, we write stat_(η,φ)ⁿ − limx = ℓ.

Lemma 3.2

Let (X, η, φ, *, ⋄) be an IFnNLS. Then for every ε > 0, u > 0 and y₁, y₂, …, y_{n − 1} ∈ X, the following statements are equivalent:

• (a) stat_(η,φ)ⁿ − limx = ℓ.

• (b) limn1n∣{k≤n:η(xk-l,y2,y3,....yn,u)≤1-ε}∣=limn1n∣{k≤n:φ(xk-l,y2,y3,....yn,u)≥ε}∣=0.

• (c) limn1n∣k≤n:η(xk-l,y2,y3,....yn,u)>1-ε and φ(xk-l,y2,y3,....yn,u)<ε∣=1

• (d) limn1n∣k≤n:η(xk-l,y2,y3,....yn,u)>1-ε∣=limn1n∣k≤n:φ(xk-l,y2,y3,....yn,u)<ε∣=1.

• (e) stat − limη(x_k − ℓ, y₂, y₃, ....y_n, u) = 1 and stat − limφ(x_k − ℓ, y₂, y₃, ....y_n, u) = 0

Theorem 3.3

Let (X, η, φ, *, ⋄) be an NA-IFnNS. If the sequence x = {x_k} in X is statistically convergent with respect to the intuitionistic fuzzy n-norm (η, φ)ⁿ, stat_(η,φ)n − limx is unique.

Theorem 3.4

Let (X, η, φ, *, ⋄) be an NA-IFnNS. If (η, φ)ⁿ − limx = ℓ then stat_(η,φ)ⁿ − limx = ℓ.

Note

Interestingly, the converse of the above theorem, which is not classically true, is true in NA-IFnNS as shown below.

Let x = {x_k} be stat_(η,φ)ⁿ that converges to ℓ ∈ X. Subsequently, for every ε > 0 and u > 0

We now prove that {x_k} is (η, φ)ⁿ - convergent to ℓ ∈ X. i.e., to prove η(x_k − ℓ, y₂, y₃, ..., y_n, u) > 1 − ε and φ(x_k − ℓ, y₂, y₃, ..., y_n, u) < ε.

We assume that η(x_k − ℓ, y₂, y₃, ..., y_n, u) ≤ 1 − ε or φ(x_k − ℓ, y₂, y₃, ..., y_n, u) ≥ ε ⇒ {k ∈ ℕ: η(x_k − ℓ, y₂, y₃, ..., y_n, u) ≤ 1 − ε or φ(x_k − ℓ, y₂, y₃, ..., y_n, u) ≥ ε} has infinitely many terms.

limn1n∣{k∈ℕ:η(xk-l,y2,y3,…,yn,u}≤1-ε or φ(x_k − ℓ, y₂, y₃, ..., y_n, u) ≥ ε}| ≠ 0. However, this finding is contradictory.

Thus, {x_k} is (η, φ)ⁿ, which converges to ℓ ∈ X.

Theorem 3.5

Let (X, η, φ, *, ⋄) be an NA-IFnNS. If stat_(η,φ)ⁿ − limx_k = x and stat_(η,φ)ⁿ − limy_k = y then stat_(η,φ)ⁿ − lim(x_k + y_k) = x + y.

Theorem 3.6

Let (X, η, φ, *, ⋄) be an NA-IFnNS over K. If lim η(x_k − ℓ, y₂, y₃, ..., y_n, t) = 1 and lim φ(x_k − ℓ, y₂, y₃, ..., y_n, t) = 1.

Definition 4.1

Let (X, η, φ, *, ⋄) be an NA-IFnNS. A sequence x = { x_n } is said to be a Cauchy sequence if, for each ε > 0 and u > 0, there exists a number n₀ ∈ ℕ such that for all n, m ≥ n₀, η( x_n − x_m, y₂, y₃, ..., y_n, u) > 1 − ε and φ( x_n − x_m, y₂, y₃, ..., y_n, u) < ε.

Definition 4.2

Let (X, η, φ, *, ⋄) be an NA-IFnNS. A sequence x = {x_k} is said to be a statistically Cauchy sequence if for every ε > 0 and u > 0, there exists ℕ such that for all k, ℓ ≥ N,

The NA-IFnNS (X, η, φ, *, ⋄) is complete if every (η, φ)ⁿ - Cauchy is (η, φ)ⁿ-convergent.

In this case, (X, η, φ, *, ⋄) is called an NA-intuitionistic fuzzy Banach space.

Theorem 4.3

In the NA-IFnNS (X, η, φ, *, ⋄) over K, every Cauchy sequence with respect to (η, φ)ⁿ is statistically Cauchy.

Theorem 4.4

Let (X, η, φ, *, ⋄) be an NA-IFnNS over K. A sequence was considered statistically significant if it converged.

Definition 5.1

A NA-IFnNS (X, η, φ, *, ⋄) over K is said to be statistically complete if every statistically Cauchy sequence with respect to (η, φ)ⁿ is statistically convergent with respect to (η, φ)ⁿ.

Theorem 5.2

Every NA-IFnNS (X, η, φ, *, ⋄) over K is statistically complete with respect to (η, φ)ⁿ.

Definition 6.1

Let (X, η, φ, *, ⋄) be a NA-IFnNS over K. A map f : X → X is called (η, φ)ⁿ continuous at point x ∈ X if the convergence of the sequence in the NA-IFnNS implies the convergence of f(x_n) to f(x) in the NA-IFnNS.

Definition 6.2

Let (X, η, φ, *, ⋄) be an NA-IFnNS over K. A map f : X → X is called statistically continuous at point x ∈ X if stat_(η,φ)ⁿ − limx_n = x, implying that stat_{(η,φ) ⁿ} − limf(x_n) = f(x).

Theorem 6.3

Let (X, η, φ, *, ⋄) be an NA-IFnNS over K. Subsequently, f is statistically continuous if f : X → X is continuous with respect to (η, φ)ⁿ.

In this study, we developed the concept of statistical convergence and statistical Cauchy sequence in the NA-IFnNS and established new results. We have proven that some properties of statistical convergence, which are not classically true, are true in the NA-IFnNS. Definition 5.1 and Definition 6.2 gives new ideas for studying completeness and continuity, respectively, in terms of statistical convergence.