Ideal convergence and ideal Cauchy sequences in intuitionistic fuzzy metric spaces

. The present study introduces the concepts of ideal convergence ( I –convergence), ideal Cauchy ( I –Cauchy) sequences, I ∗ –convergence, and I ∗ –Cauchy sequences in intuitionistic fuzzy metric spaces. It defines I –limit and I –cluster points as a sequence in these spaces. Afterward, it examines some of their basic properties. Lastly, the paper discusses whether phenomena should be further investigated.


Introduction
Based on the concept of density of positive natural numbers, statistical convergence was independently defined by Fast [8] and Steinhaus [9] in 1951. Adopting an ideal I of some subsets of the set of positive integers, Kostryko et al. [18] have characterized ideal convergence (I-convergence) as a generalization of ordinary and statistical convergence and also conceptualized the I * -convergence closely related to I-convergence. Besides, Dems [13] has extended the statistical Cauchy sequence [10] to ideals and introduced ideal Cauchy (I-Cauchy) sequences. Nabiyev et al. [3] have proposed I * -Cauchy sequences and investigated the relationship between these sequences and I-Cauchy sequences.
Fuzzy sets, defined by Zadeh [15] in 1965, have been used in many fields, such as artificial intelligence, decision-making, image analysis, probability theory, and weather forecasting. In particular, Kramosil and Michalek [12] and Kaleva and Seikkala [17] have first examined the concept of fuzzy metric spaces (FMSs). Furthermore, George and Veeramani [2], using continuous t-norms, extensively revised the concept of fuzzy metric space originally proposed by Kramosil. As a result, they established a Hausdorff topology for fuzzy metric spaces and have introduced significant advancements in this field.

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Ideal convergence and ideal Cauchy sequences in IFMSs Lately, Mihet [6] has studied the notion of point convergence (p-convergence), a weaker concept than ordinary convergence. Moreover, Gregori et al. [20] have suggested the s-convergence. Morillas and Sapena have defined the concept of standard convergence (std-convergence) [19]. Gregori and Miñana [21] have introduced the strong convergence (st-convergence), a stronger concept than ordinary convergence. Li et al. [5] have propounded the statistical convergence and statistical Cauchy sequence in FMSs and have examined some of their basic properties.
In 1986, Atanasov [14] generalized a fuzzy set introduced by Zadeh [15], accepting the membership as a fuzzy logic value rather than a single truth value, and introduced the Intuitionistic Fuzzy Set (IFS). Later, in 2004, Park [11] generalized the notion of fuzzy metric spaces to the intuitionistic fuzzy metric spaces (IFMSs) with the help of an intuitionistic set. Many studies, such as fixed point theory [16] and convergence types [1], have been studied and introduced in IFMSs. One of these studies, the statistical convergence in IFMSs, was dealt with by Varol in 2022 [4].
The current paper can be summarized in the following way. Section 2 presents some basic definitions and properties required in the following sections. Section 3 proposes the concepts of I and I * -convergence, I and I * -Cauchy sequence in IFMSs and suggests some of their basic properties. Section 4 defines the notions of I-limit points and I-cluster points of a sequence in IFMSs. The final section discusses the need for further research.

Preliminaries
This section presents the exhaustive definitions, basic properties, and theorems for ideal convergence, ideal Cauchy sequences, IFMSs and statistical convergence in IFMSs.
Example 1 ( [7]). According to the previous two definitions, the following operators are basic examples of t-norm and t-conorms, respectively.
With the help of definition 1 and 2; Park [11] has recently introduced the IFMS as follows.
Park [11] introduced a comprehensive definition of convergence of sequence in IFMSs as below.

Definition 7 ([4]
). Let (X, µ, ν, •, ▽) be an IFMS. Then, a sequence (x n ) is called statistically Cauchy sequence, if for all ε ∈ (0, 1) and u > 0, there exists N ∈ N such that An interesting generalization of statistical convergence was introduced by Kostryko et al. [18] with the help of an admissible ideal I of subsets of N, the set of positive integers. Next, we recall the basic terminology used by the authors to define this new type of convergence.
Here, ∆ denotes the symmetric difference. It must be noted that R i ∈ I.
Definition 11 ([18]). Let X be a non-empty set. A family of subsets ∅ ̸ = F ⊆ P (X) is referred to as a filter in X, ). Let I ⊆ P (N) be an admissible ideal with the condition (AP), (P i ) be a countable collection of subsets of N, and (P i ) ∈ F (I). Then, there exists a set P ⊂ N such that P ∈ F (I) and for all i, P \ P i is finite. Here, if I is an admissible ideal, then convergence in the ordinary sense implies I-convergence. Definition 13 ([18]). Let I be a non-trivial ideal in N. A sequence (x n ) is referred to as I * -convergent to x 0 ∈ R, if there exists a set 3. µ ν I-convergence and µ ν I-Cauchy sequences This section defines the concepts of ideal convergence and ideal Cauchy sequences in IFMSs. In addition, it provides some of basic properties.
Definition 16. Let I non-trivial ideal in N and (X, µ, ν, •, ▽) be an IFMS. Then, a sequence (x n ) in X is said to be ideal convergent to x 0 ∈ X, if for all u > 0 and ε ∈ (0, 1), The number x 0 is called µ ν I-limit of the sequence (x n ).  Proof. Let x n µ ν − → x 0 and I is an admissible ideal. In this case, for all u > 0 and ε ∈ (0, 1), there exists a positive integer n 0 such that n ≥ n 0 implies µ(x n , x 0 , u) > 1 − ε and ν(x n , x 0 , u) < ε, Since the set of K is finite and I is an admissible ideal, K ∈ I. Hence, Next, we shall explore the compatibility of ideal convergence with various convergence axioms. Presented below are the widely recognized axioms of classical convergence: I A constant sequence (x 0 , x 0 , . . . , x 0 , . . . ) converges to x 0 ; II The limit of a convergent sequence is unique; III Every subsequence of the converged sequence is convergent and has the same limit.
From this µ(x 1 , x 2 , u) = 1 and ν(x 1 , x 2 , u) = 0 which is a contradiction to x 1 ̸ = x 2 . (2) Suppose that an infinite set A = {n 1 < n 2 < · · · < n k < · · · } ⊆ N belongs to I. Put The set B is infinite because in the opposite case N would belong to I. Define the sequence (x n ) as follows Obviously µ ν I − lim n→∞ x n = x 0 . In addition, the sequence (x m k ) of (x n ) is constant and thus µ ν I − lim m k →∞ x m k = x 1 (see proposition (I)).
Hence, µ ν I-convergence does not satisfy the proposition (III). □ Definition 17. Let I be an admissible ideal in N and (X, µ, ν, •, ▽) be an IFMS. Then, a sequence (x n ) in X is said to be µ ν I-Cauchy sequence, if for all u > 0 and ε ∈ (0, 1), there exists an integer N ∈ N such that Theorem 2. Let I be an admissible ideal in N, (X, µ, ν, •, ▽) be an IFMS and (x n ) is a sequence in X. If the sequence (x n ) is a µ ν I-convergent sequence in X, then it is µ ν I-Cauchy sequence in X.
Then, for all u > 0 and ε ∈ (0, 1), we have Because of the definition of an admissible ideal, there exists an N / ∈ A(u, ε). Assume that
Definition 18. Let (X, µ, ν, •, ▽) be an IFMS. Then, a sequence (x n ) in X is said to be I * -convergent to x 0 ∈ X, if there exists a subset H = {h 1 < h 2 < · · · } ∈ F (I) such that The element x 0 is called the I * -limit of the sequence (x n ) and we write Theorem 3. Let (X, µ, ν, •, ▽) be an IFMS and (x n ) be a sequence in X. If Proof. By hypothesis, there is a set K ∈ I such that (1) holds, where Let u > 0 and ε ∈ (0, 1). By (1), there is a k 0 ∈ N, such that µ(x n , x 0 , u) > 1 − ε and ν(x n , x 0 , u) < ε for n > k 0 . Put
Example 6. Assume that (R, |.|) denotes the space of real numbers with the usual metric, and let a • b = ab, a ▽ b = min{a + b, 1} for all a, b ∈ [0, 1]. Define µ and ν by for all x 1 , x 2 ∈ R and u > 0. Put I = K (see Example 4). Suppose that x 0 is accumulation point of R. Hence, there exists a sequence (x n ) in R such that µ ν − lim n→∞ x n = x 0 . Define y n := x j , if n ∈ T j , j = 1, 2, . . . ; 0, otherwise.
Consequently, each of the following sets P k ∈ I (k = 1, 2, . . . ) Obviously P i ∩ P j = ∅ for i ̸ = j. Since I satisfies (AP), there exist sets R j ⊆ N such that P j ∆R j is a finite set (j = 1, 2, . . . ) It suffices to prove that Let λ ∈ (0, 1) and u > 0. Choose a m ∈ N such that 1 m < λ. Then, The set on right-hand side belongs to I by the additivity of I. Since P j ∆R j is finite (j = 1, 2, . . . ), there is an n ε ∈ N such that m+1 j=1 R j ∩ (n ε , ∞) = m+1 j=1 P j ∩ (n ε , ∞).
Hence, (2) holds. (2) Suppose x 0 ∈ X is an accumulation point of X. Then, there exists a sequence (y n ) of distinct elements of X such that y n ̸ = x 0 for any n, and µ ν − lim n→∞ y n = x 0 . Let {P 1 , P 2 , . . . } be a disjoint family of nonempty sets in I. Define a sequence (x k ) in the following way: x k = y n if k ∈ P j and x k = x 0 if k / ∈ P j , for all j. Let η ∈ (0, 1) and u > 0. Choose n ∈ N such that 1 n < η. Then, Hence, A(u, η) ∈ I and µ ν I − lim k→∞ x k = x 0 . By virtue of our assumption, we have µ ν I * − lim k→∞ x k = x 0 . Therefore, there exists a set R ∈ I such that H = N \ R ∈ F (I) and and thus ∞ j=1 R j ∈ I. Since (4), for all η ∈ (0, 1) and u > 0, Ideal convergence and ideal Cauchy sequences in IFMSs and B is finite. Since (3), H ∩ P j is finite. In addition, and P j ∆R j is finite. This proves that ideal I has the property (AP).
□ Theorem 5. Let I be an admissible ideal in N and X be an IFMS. If X has no accumulation point, then µ ν I-convergence and µ ν I * -convergence are the same.
Theorem 6. If a sequence (x n ) is an µ ν I * -Cauchy sequence, then it is µ ν I-Cauchy, for all I is an admissible ideal in N.