Double sequences with ideal convergence in fuzzy metric spaces

1.   Introduction

Statistical convergence, built on the density of natural numbers, was independently defined by Steinhaus ^[1] and Fast ^[2] in 1951. Statistical convergence as a summability method was also proposed by Schoenberg ^[3]. Since its introduction, statistical convergence has been applied in a great diversity of fields, including summability theory ^[4], locally convex sequence spaces ^[5], trigonometric series ^[6], number theory ^[7] and measurement theory ^[8].

Statistical convergence is associated with the natural density of positive integer sets, while sets with a natural density of zero represent an ideal. Building on this idea, Kostryko et al. ^[9] introduced "ideal convergence" in 2000, which generalizes statistical convergence.

They have also investigated the closely related concept of $I^*$-convergence and the condition (AP). Later, Dems ^[10] extended statistical Cauchy sequences ^[11] to ideals and proposed ideal Cauchy ($I$-Cauchy) sequences. Nabiyev et al. ^[12] further introduced $I^*$-Cauchy sequences and analyzed the relationship between current sequences and $I$-Cauchy sequences. In 2003, Mursaleen and Edely ^[13] studied statistical convergence for double sequences. Moreover, Triphaty and Triphaty ^[14] coined and characterized $I$-convergence and $I$-Cauchy sequences for double sequences. Afterward, Kumar ^[15] developed $I$ and $I^*$-convergence for double sequences, providing a more straightforward method to prove results for $I$ and $I^*$-convergence. In 2008, Das et al. ^[16] presented $I$ and $I^*$-convergence of double sequences in a metric space by exemplifying the relationships between them. Subsequently, some results on $I$-convergence of double sequences were presented in ^[17,18].

Fuzzy sets were first introduced by Zadeh ^[19] and have since been utilized by many mathematicians in topology and analysis. Fuzzy metric spaces (FMSs) extend the notion of metric spaces by introducing degrees of membership or fuzziness of points. Kramosil and Michalek ^[20] and Kaleva and Seikkala ^[21] were among the first to investigate FMSs. Building on Kramosil and Michalek's ^[20] work, George and Veeramani ^[22] redefined the concept of FMSs by utilizing a continuous t-norm and obtained the Hausdorff topology of these spaces.

Recently, Mihet ^[23] has studied the concept of point convergence ($p$-convergence), a weaker concept than ordinary convergence. Additionally, Gregori et al. ^[24] proposed the concept of $s$-convergence. Standard convergence (std-convergence) was presented by Morillas and Sapena ^[25]. Gregori and Miňana ^[26] have proposed strong convergence (st-convergence), which is a stronger concept than ordinary convergence. Statistical convergence and statistical Cauchy sequences in FMSs were proposed by Li et al. ^[27] and they have examined some of their basic properties. Moreover, Savaş ^[28] has introduced statistical convergence for double sequences in FMSs.

Inspired by previous research, we focus on ideal convergence for double sequences in FMSs. We propose $I$- and $I^*$-convergence and $I$- and $I^*$-Cauchy sequences for double sequences in FMSs and investigate some of their basic properties. We define $I$-limit points and $I$-cluster points of a double sequence in FMSs.

Our study is significant in that it provides a new approach to studying the convergence behavior of double sequences in FMSs. Ideal convergence has not been analyzed in this context, and our research contributes to filling this gap in the literature. Moreover, our results can be useful for applications in various fields.

The present paper can be summarized as follows: In Section 2 of our article, we present basic definitions and properties that are essential for the following sections. We also provide necessary background information about FMSs and the notion of convergence. In Section 3, we define $I$- and $I^*$-convergence for double sequences in FMSs, and $I$- and $I^*$-Cauchy sequences. We investigate some of their basic features, such as other types of convergence and their relations with Cauchy sequences. In Section 4, we introduce $I$-limit points and $I$-cluster points of double sequences in FMSs. Finally, in the concluding section, we summarize our findings and discuss the need for further research in this area.

2.   Preliminaries

This part thoroughly introduces the fundamental concepts, definitions and properties required to understand FMSs and convergence fully.

Definition 2.1. ^[21] Let $\circ : [0, 1]^2 \to [0, 1]$ be a binary operation. We say that $\circ$ is a triangular norm (t-norm) if it satisfies the following conditions:

(1) $\circ$ is both associative and commutative;

(2) $t\circ 1 = t$ for all $t\in [0, 1]$;

(3) Whenever $t_1\leq t_3$ and $t_2\leq t_4$ for each $t_1, t_2, t_3, t_4\in [0, 1]$, it holds that $t_1\circ t_3\leq t_2\circ t_4$.

Example 2.2. ^[19] According to the above definition, the following operators are a t-norm:

(1) $\sigma\circ \tau = \sigma\tau$,

(2) $\sigma\circ \tau = \min\{\sigma, \tau\}$.

Definition 2.3. ^[20] Let $\vartheta$ be a fuzzy set on $\mathbb{X}^2 \times (0, \infty)$, where $\mathbb{X}$ is an arbitrary set and $\circ$ be a continuous t-norm. If the following requirements must be fulfilled for all an $u, v > 0$ and $x_1, x_2, x_3\in \mathbb{X}$, then $\vartheta$ is referred to as a fuzzy metric on $\mathbb{X}$,

(1) $\vartheta(x_1, x_2, u) > 0$;

(2) $\vartheta(x_1, x_2, u) = 1 \Leftrightarrow x_1 = x_2$;

(3) $\vartheta(x_1, x_2, u) = \vartheta(x_2, x_1, u)$;

(4) $\vartheta(x_1, x_3, u+v)\geq \vartheta(x_1, x_2, u)\circ \vartheta(x_2, x_3, v)$;

(5) The function $\vartheta_{x_1x_2} : (0, \infty) \to [0, 1]$, defined by $\vartheta_{x_1x_2}(u) = \vartheta(x_1, x_2, u)$ is continuous.

The 3-tuple $(\mathbb{X}, \vartheta, \circ)$ is called fuzzy metric space.

Example 2.4. ^[20] Consider the set $\mathbb{X} = \mathbb{R}$ and define the binary operation $\circ$ as $\sigma \circ \tau = \sigma \tau$. Additionally, we define the fuzzy set $\vartheta$ as follows:

Therefore, we can conclude that $(\mathbb{X}, \vartheta, \circ)$ is an FMS.

Definition 2.5. ^[22] Let $(\mathbb{X}, \vartheta, \circ)$ be an FMS. If $a\in \mathbb{X}$, then the open ball centered at $a$ with radius $\varepsilon, 0 < \varepsilon < 1$ is the set of points $x\in \mathbb{X}$ contained in

Definition 2.6. ^[28] A double sequence $(x_{jk})$ in $\mathbb{X}$ is said to be convergent to $x_0\in \mathbb{X}$ with respect to fuzzy metric $\vartheta$ if, for all $\varepsilon \in (0, 1)$ and $u > 0$, there exists $N_\varepsilon \in \mathbb{N}$ such that $j, k\geq N_\varepsilon$ implies

or equivalently

and is denoted by $\vartheta-\lim\limits_{j, k \to \infty} x_{jk} = x_0$ or $x_{jk}\xrightarrow{\vartheta}x_0$ as $j, k\to \infty$.

Definition 2.7. ^[13] Let $E\subseteq \mathbb{N}^{2}$ and $E_{mn} = \{(j, k)\in E : j \leq m \text{ and } k\leq n\}$. The set $E$ is said to have double natural density, denoted by $\delta_2(E)$, is defined as:

if the limit exists. It can be observed that if the set $E$ is finite, then $\delta_2(E) = 0$.

Definition 2.8. ^[28] A double sequence $(x_{jk})$ in $\mathbb{X}$ is referred to as statistically convergent to $x_0\in \mathbb{X}$ with respect to fuzzy metric $\vartheta$ if, for all $\varepsilon \in (0, 1)$ and $u > 0$,

and is denoted by $st_2-\lim\limits_{j, k \to \infty} x_{jk} = x_0$.

The definitions of ideal and filter are provided below to explain the main results regarding ideal convergence. Then, the concepts needed in the sequel of the study are mentioned.

Definition 2.9. ^[16] Let $\mathbb{X}\neq\emptyset$. An ideal on $\mathbb{X}$ is a collection of subsets of $\mathbb{X}$ such that

(a) The empty set $\emptyset$ is an element of $I$.

(b) If $\mathcal{U}$ and $\mathcal{V}$ are set in $I$, then their union $\mathcal{U}\cup \mathcal{V}$ is also an element of $I$.

(c) If $\mathcal{U}$ is a set in $I$ and $\mathcal{V}$ is a subset of $\mathbb{X}$ such that $\mathcal{V}\subseteq \mathcal{U}$, then $\mathcal{V}$ is an element of $I$.

If $\mathbb{X}\notin I$ and $I\neq \emptyset$, then $I$ is called a non-trivial ideal. Additionally, if $I$ is a non-trivial ideal in $\mathbb{X}$ and

then $I$ is referred to as an admissible ideal.

In the current study, $I_2$ denotes a non-trivial admissible ideal of $\mathbb{N}^2$.

Definition 2.10. ^[16] A strongly admissible ideal on $I_2$ is a collection of subsets of ${\mathbb{N}}^2$ such that

(a) ${\{r\}\times \mathbb{N}}\in I_2$,

(b) ${\mathbb{N}\times \{r\}}\in I_2$.

For example, ${I^0_2} = \{P\subseteq \mathbb{N}^{2} : (\exists l(P) \in \mathbb{N})(j, k\geq l(P) \Rightarrow (j, k)\notin P)\}$ is a non-trivial strongly admissible ideal. Any strongly admissible ideal is an admissible ideal.

Definition 2.11. ^[16] Let $I_2\subseteq 2^{\mathbb{N}^{2}}$ be an admissible ideal, ($P_i$) be a sequence of mutually disjoint sets of $I_2$ and ($R_i$) be a subset of $\mathbb{N}^{2}$. Then, $I_2$ satisfies the condition (AP2) if, for all ($P_i$), there is a sequence ($R_i$) such that, for all $i\in \mathbb{N}$, $P_i \Delta R_i \in I^0_2$ i.e., $P_i \Delta R_i$ is included in limited quantities union of rows and columns in $\mathbb{N}^{2}$ and $R = \bigcup\limits_{i}R_i \in I_2$. Here, $\Delta$ denotes the symmetric difference. Note that $R_i\in I_2$.

Definition 2.12. ^[16] Let $\mathbb{X}\neq\emptyset$. A filter on $\mathbb{X}$ is a collection of subsets of $\mathbb{X}$ such that

(a) The empty set $\emptyset$ is not an element of $\mathcal{F}$.

(b) If $\mathcal{U}$ and $\mathcal{V}$ are sets in $\mathcal{F}$, then their intersection $\mathcal{U}\cap \mathcal{V}$ is also an element of $\mathcal{F}$.

(c) If $\mathcal{V}$ is a set in $\mathcal{F}$ and $\mathcal{U}$ is a subset of $\mathbb{X}$ such that $\mathcal{V}\subseteq \mathcal{U}$, then $\mathcal{U}$ is an element of $\mathcal{F}$.

Furthermore, let $I_2$ be a non-trivial ideal. Then the collection $\mathcal{F}(I_2) = \{\mathbb{N}^{2}\setminus S: S\in I_2\}$ is a filter on $\mathbb{N}^{2}$ and is referred to as the filter associated with the ideal $I_2$.

Proposition 2.13. ^[17] Let $(P_i)$ be a countable collection of subsets of $\mathbb{N}^{2}$ such that $(P_i) \in \mathcal{F}\left(I_2\right)$, for all $i$, where $\mathcal{F}\left(I_2\right)$ is a filter associated with a strongly admissible ideal $I_2$ with the property (AP2). Then, there exists a set $P$ that belongs to the filter $\mathcal{F}(I_2)$ and has the property that the set of elements in $P$ that do not belong to $P_i$ is finite for every index $i$.

Definition 2.14. ^[16] Let $I_2\subset 2^{\mathbb{N}^{2}}$ be a non-trivial ideal. A double sequence $(x_{jk})$ in a metric space $(\mathbb{X}, \rho)$ is called ideal convergent ($I_2$-convergent) to $x_0\in \mathbb{X}$, written as $I_2-\lim\limits_{j, k\to \infty} x_{jk} = x_0$ or $x_{jk} \xrightarrow{I_2}x_0$ as $j, k\to \infty$ if, for all $\varepsilon > 0$,

If we choose $I_2 = \left\{S\subseteq \mathbb{N}^2 : S \text{ is of the form } (\mathbb{N}\times A) \cup (A\times\mathbb{N}) \right\}$, where $A$ is a finite subset of $\mathbb{N}$, then $I_2$-convergent coincides with ordinary convergence of double sequences.

If we choose $I_2 = \left\{S\subseteq \mathbb{N}^2 :\delta_2(S) = 0 \right\}$, then $I_2$-convergent is equivalent to the statistical convergence of double sequences.

Definition 2.15. ^[16] A double sequence $(x_{jk})$ in a metric space $(\mathbb{X}, \rho)$ is referred to as $I_2^{*}$-convergent to $x_0\in \mathbb{X}$, where $I_2\subset 2^{\mathbb{N}^{2}}$ be a non-trivial ideal if exists a set

such that

We abbreviate it as $I_2^{*}-\lim\limits_{j, k\to \infty} x_{jk} = x_0$ or $x_{jk} \xrightarrow{I_2^{*}}x_0$.

Definition 2.16. ^[10] A double sequence $(x_{jk})$ in a metric space $(\mathbb{X}, \rho)$ is referred to as an ideal Cauchy ($I_2$-Cauchy) sequence in $\mathbb{X}$, where $I_2\subseteq 2^{\mathbb{N}^{2}}$ be a strongly admissible ideal if, for all $\varepsilon > 0$, there exists an $(p, q)\in \mathbb{N}^{2}$ such that

Definition 2.17. ^[17] Let $I_2$ be a strongly admissible ideal in $\mathbb{N}^{2}$. A double sequence $(x_{jk})$ in a metric space $(\mathbb{X}, \rho)$ is called an $I^*_2$-Cauchy sequence in $\mathbb{X}$ if there exists a set

such that

Definition 2.18. ^[18] Let $(\mathbb{X}, \rho)$ be a metric space and $(x_{jk})$ be a double sequence in $\mathbb{X}$. Then, an element $x_0\in \mathbb{X}$ is referred to as an $I_2$-limit point of $(x_{jk})$ if there is a set

and

Definition 2.19. ^[18] Let $(\mathbb{X}, \rho)$ be a metric space and $(x_{jk})$ be a double sequence in $\mathbb{X}$. Then, an element $x_0\in \mathbb{X}$ is called an $I_2$-cluster point of $(x_{jk})$ if, for all $\varepsilon > 0$, $\{(j, k) \in \mathbb{N}^{2} : \rho(x_{jk}, x_0)\leq \varepsilon\}\notin I_2$.

The set of all $I_2$-limit points and $I_2$-cluster points of a double sequence $x$ are denoted by $I_2(\Lambda_x)$ and $I_2(\Gamma_x)$, respectively.

3.   $\vartheta(I_2)$-convergence and $\vartheta(I_2)$-Cauchy sequence

In this chapter, we define the notions of ideal convergence and ideal Cauchy sequences for double sequences in FMSs and discuss some of their basic properties. Throughout this chapter, for brevity, we shall often write $\mathbb{X}$ instead of "$(\mathbb{X}, \vartheta, \circ)$" and $(x_{jk})$ instead of a "double sequence $(x_{jk})$".

Definition 3.1. Let $I_2 \subseteq 2^{\mathbb{N}^{2}}$ be a non-trivial ideal. A sequence $(x_{jk})$ is referred to as $\vartheta(I_2)$-convergent to $x_0\in \mathbb{X}$ if, for all $u > 0$ and $\varepsilon\in (0, 1)$,

and is denoted by $\vartheta(I_2)-\lim\limits_{j, k\to \infty} x_{jk} = x_0$ or $x_{jk} \xrightarrow{\vartheta(I_2)} x_0$ as $j, k\to \infty$. The number $x_0$ is called the $I_2$-limit of $(x_{jk})$.

Example 3.2. If we choose $I_2 = I_0$ and $I_2 = I_\delta = \{A\subseteq \mathbb{N}^{2} : \delta_2(A) = 0\}$, then $I_2$-convergence is the same as ordinary convergence and statistical convergence, respectively.

The following theorem presents, well-known in ordinary convergence, which gives whether the following expressions satisfy at ideal convergence:

Ⅰ. Every constant double sequence converges to yourself.

Ⅱ. The limit of converged double sequences can be determined by uniquely.

Ⅲ. Every subsequence of the converged double sequence is convergent and has the same limit.

Theorem 3.3. Let $I_2\subseteq 2^ {\mathbb{N}^{2}}$.

(1) The $I_2$-convergence satisfies (Ⅰ) and (Ⅱ).

(2) Every subsequence of an $I_2$-convergent sequence is not $I_2$-convergent if $I_2$ is a strongly admissible ideal and contains an infinite set.

Proof.

(1) It is clear that $\vartheta(I_2)$-convergence satisfies proposition (Ⅰ). We prove that it satisfies (Ⅱ) as well. Suppose that $\vartheta(I_2)-\lim\limits_{j, k\to \infty} x_{jk} = x_0$, $\vartheta(I_2)-\lim\limits_{j, k\to \infty}x_{jk} = x_1$ and $x_0\neq x_1$. Then, by assumption and Remark 2.12, the sets

and

are elements of $\mathcal{F}(I_2)$. Hence, the set $K = (\mathbb{N}^{2}\setminus A)\cap (\mathbb{N}^{2} \setminus B)$ is an element of $\mathcal{F}(I_2)$. Choose $u > 0$ and $\varepsilon = \frac{1}{n}$, $(n = 2, 3, \ldots)$. Thus, there exists a $(t, s)\in K$ such that

From this $\vartheta(x_0, x_1, u) = 1$ which is a contradiction to $x_0\neq x_1$.

(2) Suppose that an infinite set $A = \{(j_t, k_t)\in \mathbb{N}^{2} : j_1 < j_2 < \ldots < j_t < \ldots; k_1 < k_2 < \ldots < k_t < \ldots\} \subseteq \mathbb{N}^{2}$ belongs to $I_2$. We put

The set $\mathbb{N}^{2}\setminus A$ is infinite because in the opposite case $\mathbb{N}^{2}$ would belong to $I_2$. Define the sequence $(x_{jk})$ as follows

Obviously $\vartheta(I_2)-\lim\limits_{j, k\to \infty} x_{jk} = x_1$. In addition, the subsequence $y = (x_{j_tk_t})$ of $(x_{jk})$ is stationary and thus ${\vartheta(I_2)-\lim y = x_0 }$ (see proposition (Ⅰ)). Hence, $I_2$-convergence does not satisfy the proposition (Ⅲ).

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Proposition 3.4. Let $I_2$ be a non-trivial ideal given that

then $\lim\limits_{j, k \to \infty}\vartheta(x_{jk}, x_0, u) = 1$ implies $\vartheta(I_2)-\lim\limits_{j, k \to \infty}x_{jk} = x_0$.

Proof. Suppose that $I_2$ be a non-trivial ideal such that

and $\lim\limits_{j, k \to \infty}\vartheta(x_{jk}, x_0, u) = 1$. Let $u > 0$ and $\varepsilon\in (0, 1)$ be given. Since $(x_{jk})$ is convergent to $x_0 \in \mathbb{X}$, then there exists a $N_\varepsilon \in \mathbb{N}$ such that $\vartheta(x_{jk}, x_0, u) > 1-\varepsilon$ whenever $j, k \geq N_\varepsilon$. Hence, there exists a set

where $\mathcal{U} = \mathbb{N} \times\{1, 2, 3 \ldots N_\varepsilon-1\}$ and $\mathcal{V} = \{1, 2, 3 \ldots N_\varepsilon-1\} \times \mathbb{N}$. From the hypothesis $\mathcal{U}\cup \mathcal{V} \in I_2$. Since $I_2$ is an ideal, then $P \in I_2$. Consequently, $\vartheta(I_2)-\lim\limits_{j, k \rightarrow \infty} x_{jk} = x_0$. □

Definition 3.5. A sequence $(x_{jk})$ is referred to as Cauchy sequence in $\mathbb{X}$ if, for all $u > 0$ and $\varepsilon \in (0, 1)$, exists an integer $N_\varepsilon \in \mathbb{N}$ such that

whenever $j, k, p, q \geq N_\varepsilon$ or equivalently

Definition 3.6. A sequence $(x_{jk})$ is said to be $\vartheta(I_2)$-Cauchy sequence in $\mathbb{X}$, where $I_2$ is a strongly admissible ideal if, for all $u > 0$ and $\varepsilon\in (0, 1)$, there exists an integer $(p, q)\in \mathbb{N}^{2}$ such that

Proposition 3.7. Let $I_2$ be a strongly admissible ideal in $\mathbb{N}^{2}$. If $(x_{jk})$ is a Cauchy sequence in $\mathbb{X}$, then it is a $\vartheta(I_2)$-Cauchy sequence.

Proof. Let $u > 0$ and $\varepsilon\in (0, 1)$ be given. Since $(x_{jk})$ is Cauchy sequence in $\mathbb{X}$, for all $j, k, p, q \geq N_\varepsilon$, there exists an integer $N_\varepsilon \in \mathbb{N}$ such that $\vartheta(x_{jk}, x_{pq}, u) > 1-\varepsilon$. Hence, there exists a set

where $\mathcal{U} = \mathbb{N} \times\{1, 2, 3 \ldots N_\varepsilon-1\}$ and $\mathcal{V} = \{1, 2, 3 \ldots N_\varepsilon-1\} \times \mathbb{N}$. Since $I_2$ is a strongly admissible ideal, $\mathcal{U}\cup \mathcal{V} \in I_2$. Therefore $P \in I_2$. Consequently, $(x_{jk})$ is a $\vartheta(I_2)$-Cauchy sequence in $\mathbb{X}$. □

Theorem 3.8. For any double sequence, $\vartheta(I_2)$-convergent implies $\vartheta(I_2)$-Cauchy sequence if $I_2$ is a strongly admissible ideal in $\mathbb{N}^{2}$.

Proof. Let $\vartheta(I_2)-\lim\limits_{j, k\to \infty}x_{jk} = x_0$. Then, for all $u > 0$ and $\varepsilon\in (0, 1)$, we have

Because of the definition of a strongly admissible ideal, there exists a $(p, q)\notin A(u, \varepsilon)$. Assume that

Considering the following inequality

we observe that if $(j, k)\in B$, then

Moreover, we have $\vartheta(x_{pq}, x_0, u) > 1-\varepsilon$ because $(p, q)\notin A(u, \varepsilon)$. Hence, $\vartheta(x_{jk}, x_0, u)\leq 1-\varepsilon$, then $(j, k)\in A(u, \varepsilon)$. In this case, for all $u > 0$ and $\varepsilon\in (0, 1)$, $B\subseteq A(u, \varepsilon) \in I_2$. Consequently, $(x_{jk})$ is a $\vartheta(I_2)$-Cauchy sequence. □

Definition 3.9. A sequence $(x_{jk})$ is referred to as $\vartheta(I_2^*)$-convergent to $x_0\in \mathbb{X}$ if there exists a set

such that

and is denoted by $\vartheta(I_2^*)-\lim\limits_{j, k \to \infty} x_{jk} = x_0$ or $x_{jk} \xrightarrow{\vartheta(I_2^*)} x_0$ as $j, k\to \infty$.

Theorem 3.10. $\vartheta(I_2^*)-\lim\limits_{j, k \to \infty} x_{jk} = x_0$ implies ${\vartheta(I_2)- \lim\limits_{j, k \to \infty} x_{jk} = x_0}$ that if $I_2$ is a strongly admissible ideal in $\mathbb{N}^{2}$.

Proof. By hypothesis, there is a set $K\in I_2$ such that (3.1) holds, where

Let $u > 0$ and $\varepsilon\in (0, 1)$. Then, there exists an integer $n_0 \in \mathbb{N}$ such that $\vartheta(x_{j_pk_p}, x_0, u) > 1-\varepsilon$ for $j_p, k_p > n_0$. Hence,

where $B = \left\{1, 2, \ldots, \left(n_0-1\right)\right\}$. Since $K \cup\left(H \cap\left(\left(B \times \mathbb{N}\right) \cup\left(\mathbb{N} \times B \right)\right)\right) \in I_2$, then $A(u, \varepsilon) \in I_2$. As a result, $\vartheta(I_2)-\lim\limits_{j, k\to \infty}x_{jk} = x_0$. □

The following Example 3.11 states that the converse of Theorem 3.10 does not always hold.

Example 3.11. Let $(\mathbb{R}, |.|)$ be a metric space and $x\circ y = xy$, for all $x, y\in [0, 1]$. If, for every $x, y\in\mathbb{R}$ and $s > 0$,

then $(\mathbb{R}, \vartheta, \circ)$ is an FMS. Let $\Delta_j = \{(m, n): \min\{m, n\}\in K_j\}$ such that $K_j = \{2^{j-1}(2s-1): s = 1, 2, ...\}$ be a decomposition of $\mathbb{N}$. Besides, $\{\Delta_j\}_{j\in\mathbb{N}}$ is a decomposition of $\mathbb{N}^2$ and

is a strongly admissible ideal. We define a sequence $(x_{st})$ by

On the other hand,

Hence, $\vartheta(I_2)-\lim\limits_{s, t\to\infty}x_{st} = 0$. However, this sequence does not $\vartheta(I^*_2)$ convergent to zero.

Theorem 3.12. Let $I_2$ be an admissible ideal, $(x_{jk})$ be a sequence in $\mathbb{X}$ and $x_0\in \mathbb{X}$.

(1) If the $I_2$ ideal has the condition (AP2), then $\vartheta(I_2)-\lim\limits_{j, k\to \infty} x_{jk} = x_0$ implies $\vartheta(I_2^*)-\lim\limits_{j, k\to \infty} x_{jk} = x_0$.

(2) If $\mathbb{X}$ has at least one accumulation point and $\vartheta(I_2)-\lim\limits_{j, k \rightarrow \infty} x_{jk} = x_0$ implies $\vartheta(I_2^*)-\lim\limits_{j, k \rightarrow \infty} x_{jk} = x_0$, then $I_2$ has the property (AP2).

Proof.

(1) Let $x_{jk}\xrightarrow{\vartheta(I)}x_0$ and $I_2$ satisfy the condition (AP2). Then, for all $u > 0$ and $\varepsilon \in (0, 1)$ the set

Put

Obviously, $P_t\cap P_s = \emptyset$ for $t\neq s$ and $P_t\in I_2$ $(t = 1, 2, \ldots)$. Since $I_2$ satisfies (AP2), there exists sets $R_s\subseteq \mathbb{N}^{2}$ such that, for all $s\in \mathbb{N}$, $P_s \Delta R_s$ is contained in limited quantities union of rows and columns in $\mathbb{N}^{2}$ and $R = \bigcup\limits_{s = 1}^{\infty}R_s \in I_2$.

It suffices to prove that

where $H = \mathbb{N}^{2}\setminus R$.

Let $\eta\in (0, 1)$ and $u > 0$. Choose $m\in \mathbb{N}$ such that $\frac{1}{m} < \eta$. Then,

The set on the right hand side belongs to $I_2$ by the additivity of $I_2$. Since, for all $s\in \mathbb{N}$, $P_s \Delta R_s$ is included in limited quantities union of rows and columns, there is an $n_0\in \mathbb{N}$ such that

If $(j, k)\notin R$ and $j, k > n_0$, then $(j, k)\notin \bigcup\limits_{s = 1}^{m+1}R_s$. Hence, $(j, k)\notin \bigcup\limits_{s = 1}^{m+1} P_s$. However,

Consequently, (3.2) holds.

(2) Suppose $x_0 \in \mathbb{X}$ is an accumulation point of $\mathbb{X}$. Then, there exists a sequence $(y_n)$ of distinct elements of $\mathbb{X}$ such that $y_n \neq x_0$ for any $n$, $\lim\limits_{n \to \infty} \vartheta(y_n, x_0, u) = 1$. Let $\left\{P_1, P_2, \ldots\right\}$ be a disjoint family of nonempty sets in $I_2$. Define a sequence $(x_{jk})$ in the following way: $x_{jk} = y_n$ if $(j, k) \in P_t$ and $x_{jk} = x_0$ if $(j, k) \notin P_t$, for all $t$. Let $\eta\in (0, 1)$ and $u > 0$. Choose $n \in \mathbb{N}$ such that $\frac{1}{n} < \eta$. Then,

Hence, $A(u, \eta) \in I_2$ and $\vartheta(I_2)-\lim\limits_{j, k \to \infty} x_{jk} = x_0$. By virtue of our assumption, we have ${\vartheta(I_2^*)-\lim\limits_{j, k \to \infty} x_{jk} = x_0}$. Therefore, there exists a set $R \in I_2$ such that $H = \mathbb{N}^{2} \setminus R \in \mathcal{F}(I_2)$ and

Put $R_t = P_t \cap R$ for $t \in \mathbb{N}$. Then, $R_t \in I_2$ for all $t \in \mathbb{N}$. Moreover, $\bigcup\limits_{t = 1}^{\infty} R_t = R \cap \bigcup\limits_{t = 1}^{\infty} P_t \subset R$ and thus $\bigcup\limits_{t = 1}^{\infty} R_t \in I_2$. Let $t$ be a fixed element in $\mathbb{N}$. Suppose the intersection $P_t \cap H$ is not contained in the limited quantities union of rows and columns in $\mathbb{N}^{2}$. In that case, there must exist an infinite sequence of elements $\left\{\left(j_n, k_n\right)\right\}$ in $H$ such that both $j_n$ and $k_n$ tend to infinity, and $x_{j_n k_n} = y_t \neq x_0$ for all $n \in \mathbb{N}$. This contradicts (3.3). Therefore, $P_t \cap H$ should be included in the limited quantities union of rows and columns in $\mathbb{N}^{2}$. Consequently, the set $P_t \Delta R_t = P_t \setminus R_t = P_t \setminus R = P_t \cap H$ is also included in the limited quantities union of rows and columns. This proves that the ideal $I_2$ satisfies property (AP2).

□

Theorem 3.13. Let $I_2$ be a strongly admissible ideal in $\mathbb{N}^{2}$. If $\mathbb{X}$ has no accumulation point, then $\vartheta(I_2)$-convergence coincides with $\vartheta(I^*_2)$-convergence.

Proof. Let $x_0\in \mathbb{X}$ and $x_{jk} \xrightarrow{\vartheta(I_2)} x_0$. Thanks to Theorem 3.10, it suffices to prove that $x_{jk} \xrightarrow{\vartheta(I_2^*)} x_0$ as $j, k\to \infty$. Since $\mathbb{X}$ has no accumulation points, there exists $u > 0$ and $\varepsilon\in (0, 1)$ such that

From the assumption $\{(j, k) \in \mathbb{N}^{2} : \vartheta(x_{jk}, x_0, u)\leq 1-\varepsilon\}\in I_2$. Hence,

and obviously $x_{jk} \xrightarrow{\vartheta(I_2^*)} x_0$. □

Theorem 3.14. Let $I_2$ satisfy the condition (AP2) and $x = (x_{jk})$ be a sequence in $\mathbb{X}$. Then, the assumptions below are equivalent:

(1) $\vartheta(I_2)-\lim\limits_{j, k\to \infty}x_{jk} = x_0$;

(2) There exist $y = (y_{jk})$, $z = (z_{jk})$ in $\mathbb{X}$ such that $x = y+z$, $\lim\limits_{j, k\to \infty}\vartheta(y_{jk}, x_0, u) = 1$ and $supp z\in I_2$, where $supp z = \{(j, k)\in \mathbb{N}^{2} : z_{jk}\neq \theta_\mathbb{X}\}$.

Proof. Assume that $\vartheta(I_2)-\lim\limits_{j, k\to \infty}x_{jk} = x_0$. In that case, by Theorem 3.12, there exists a set $H\in \mathcal{F}(I_2)$, $H = \{(j_t, k_t) \in \mathbb{N}^{2} : j_1 < j_2 < \ldots < j_t < \ldots; k_1 < k_2 < \ldots < k_t < \ldots \}$ such that $\lim\limits_{j_t, k_t\to \infty}\vartheta(x_{j_tk_t}, x_0, u) = 1$. Let us define a sequence $y = (y_{jk})$ in $\mathbb{X}$ such that

It is clear that $\lim\limits_{j, k\to \infty}\vartheta(y_{jk}, x_0, u) = 1$. Further, let $z_{jk} = x_{jk}-y_{jk}$, $(j, k)\in \mathbb{N}^{2}$. We have $\{(j_t, k_t)\in \mathbb{N}^{2} : z_{j_tk_t}\neq 0\}\in I_2$, because we have

In addition, $supp z\in I_2$ and by (3.4), we write $x = y+z$.

Now, let $y = (y_{jk})$ and $z = (z_{jk})$ be two sequences in $\mathbb{X}$. This sequences satisfy $x = y+z$, ${\lim\limits_{j, k\to \infty}\vartheta(y_{jk}, x_0, u) = 1}$ and $supp z\in I_2$. We prove that

Assume that $H = \{(j_t, k_t)\in \mathbb{N}^{2} : z_{j_tk_t} = \theta_{\mathbb{X}}\}\subset \mathbb{N}^{2}$. We have $H\in \mathcal{F}(I_2)$, because

Hence, $x_{jk} = y_{jk}$ if $(j, k)\in H$. Therefore, we achieve that there exists a set

such that

By Theorem 3.12, (3.5) is hold. □

Definition 3.15. Let $I_2$ be a strongly admissible ideal on $\mathbb{N}^{2}$. If exists a set

such that

then a sequence $(x_{jk})$ in $\mathbb{X}$ is referred to as $\vartheta(I_2^*)$-Cauchy sequence.

Theorem 3.16. Let $I_2$ be a strongly admissible ideal on $\mathbb{N}^{2}$. If a sequence $(x_{jk})$ in $\mathbb{X}$ is a $\vartheta(I_2^*)$-Cauchy sequence, then it is a $\vartheta(I_2)$-Cauchy.

Proof. Assume that $(x_{jk})$ be an $\vartheta(I_2^*)$-Cauchy sequence. In that case, there exists a set

such that $\lim\limits_{{j_t, k_t, p_t, q_t\to \infty}}\vartheta(x_{j_tk_t}, x_{p_tq_t}, u) = 1$. Hence, there exists a positive integer $n_0$ such that for $j_t, k_t, p_t, q_t > n_0$ implies $\vartheta(x_{j_tk_t}, x_{p_tq_t}, u) > 1-\varepsilon$, where $u > 0$ and $\varepsilon\in (0, 1)$. In other words,

where $B = \left\{1, 2, \ldots, \left(n_0-1\right)\right\}$. Since $K \cup\left(H \cap\left(\left(B \times \mathbb{N}\right) \cup\left(\mathbb{N} \times B \right)\right)\right) \in I_2$, then $A(u, \varepsilon) \in I_2$. Consequently, the sequence $(x_{jk})$ is a $\vartheta(I_2)$-Cauchy. □

Theorem 3.17. Let $I_2$ be a strongly admissible ideal on $\mathbb{N}^{2}$. If the $I_2$ ideal has the condition (AP2), then $\vartheta(I_2)$-Cauchy sequence and $\vartheta(I_2^*)$-Cauchy sequence coincide.

Proof. In Theorem (3.16), it was shown that while the $(x_{jk})$ sequence is the $\vartheta(I_2^*)$-Cauchy sequence, it is the $\vartheta(I_2)$-Cauchy sequence without requiring the (AP2) condition. Therefore, if the $I_2$ ideal satisfies the (AP2) condition, showing that the $\vartheta(I_2)$-Cauchy sequence is the $\vartheta(I_2^*)$-Cauchy sequence will complete the proof. Now, assume that $(x_{jk})$ be a $\vartheta(I_2)$-Cauchy sequence in $\mathbb{X}$. Then, for all $\varepsilon \in (0, 1)$ and $u > 0$, there exists ${(p(\varepsilon), q(\varepsilon)) \in \mathbb{N}^{2}}$ such that

Let

where $p_s = p(\frac{s-1}{s}), q_s = q(\frac{s-1}{s})$. It is evident that $P_s \in \mathcal{F}\left(I_2\right)$ for $s = 1, 2, \ldots$. Since $I_2$ satisfy the property (AP2), according to Proposition (2.13), there exists a set $P$ such that $P \in \mathcal{F}(I_2)$ and has the property that the set of elements in $P$ that do not belong to $P_s$ is a limited quantity for every index $s$.

Let $\varepsilon\in (0, 1)$, $u > 0$ and $m \in \mathbb{N}$ such that $m > \frac{1}{\varepsilon}$. If $(j, k), (p, q) \in P$, then $P \setminus P_s$ is a limited quantities set, implying that there exists a $n = n(m)$ such that, for all $j, k, p, q > n(m)$, $(j, k), (p, q) \in P_s$.

Hence, it follows that

for all $j, k, p, q > n(m)$. Thus, for all $\varepsilon\in (0, 1)$ and $u > 0$, there exists $n = n(\varepsilon)$ such that, for $j, k, p, q > n(\varepsilon)$ and $(j, k), (p, q) \in P$,

i.e., the sequence $(x_{jk})$ is a $\vartheta(I_2^*)$-Cauchy sequence. □

4.   $\vartheta(I_2)$-limit points and $\vartheta(I_2)$-cluster points

In the current part, we characterize $\vartheta(I_2)$-limit points and $\vartheta(I_2)$-cluster points of a double sequence in FMSs. Moreover, we analyze the connection between the concept mentioned earlier and study that the set of $\vartheta(I_2)$-cluster points are closed.

Definition 4.1. Let $I_2$ be a strongly admissible ideal on $\mathbb{N}^{2}$ and $(x_{jk})$ be a sequence in $\mathbb{X}$. An element $x_0\in \mathbb{X}$ is referred to as an $\vartheta(I_2)$-limit point of sequence $(x_{jk})$, if there exists a set

such that

Definition 4.2. Let $I_2$ be a strongly admissible ideal on $\mathbb{N}^{2}$ and $(x_{jk})$ be a sequence in $\mathbb{X}$. An element $x_0\in \mathbb{X}$ is called an $\vartheta(I_2)$-cluster point of $(x_{jk})$ if, for all $u > 0$ and $\varepsilon\in (0, 1)$,

$\vartheta(I_2)(\Lambda_x)_2$ and $\vartheta(I_2)(\Gamma_x)_2$ denote the set of all $\vartheta(I_2)$-limit points and $\vartheta(I_2)$-cluster points of a sequence $x = (x_{jk})$, respectively.

Proposition 4.3. Let $(x_{jk})$ be a sequence in $\mathbb{X}$ and $I_2$ be a strongly admissible ideal on $\mathbb{N}^{2}$. Then,

Proof. Let $x_0\in \vartheta(I_2)(\Lambda_x)_2$, then there exists a set

such that

Let $u > 0$ and $\varepsilon\in (0, 1)$. According to (4.1), there exists a $N_\varepsilon\in \mathbb{N}$ such that for $j, k > N_\varepsilon$ implies $\vartheta(x_{jk}, x_0, u) > 1-\varepsilon$. Hence,

and thus $\{(j, k)\in \mathbb{N}^{2} : \vartheta(x_{jk}, x_0, u) > 1-\varepsilon\}\notin I_2$ which means that $x_0\in \vartheta(I_2)(\Gamma_x)_2$. □

Theorem 4.4. Let $(x_{jk})$ be a sequence in $\mathbb{X}$. Then, the set $\vartheta(I_2)(\Gamma_x)_2$ is closed if $I_2$ is a strongly admissible ideal on $\mathbb{N}^{2}$.

Proof. Let $y\in \overline{\vartheta(I_2)(\Gamma_x)_2}$, $u > 0$ and $\varepsilon\in (0, 1)$. Then, there exists an $x_0\in B^{\varepsilon}_y(x)\cap \vartheta(I_2)(\Gamma_x)_2$, where $B^{\varepsilon}_y(x) = \{x\in \mathbb{X}:\vartheta(y, x, u) > 1-\varepsilon\}$. Suppose that $\delta\in (0, 1)$ such that

Hence,

Consequently, $\{(j, k)\in \mathbb{N}^{2} : \vartheta(y, x_{jk}, u) > 1-\varepsilon\}\notin I_2$ and $y\in \vartheta(I_2)(\Gamma_x)_2$. □

5.   Conclusions

We showed the ideal convergence of double sequences using the concept of fuzzy metric space in the sense of George and Veeramani ^[22]. Besides, we introduced the $\vartheta(I^*)$-convergent of double sequences and $\vartheta(I^*)$-Cauchy sequence with regards to fuzzy metric $\vartheta$ and discussed the relations between them. In addition, we proved that $\vartheta(I_2)$-convergence and $\vartheta(I^*_2)$-convergence are equivalent for an $I_2$ ideal with the condition (AP2). Lastly, we defined $\vartheta(I_2)$-limit and $\vartheta(I_2)$-cluster points of a double sequence and showed every $\vartheta(I_2)$-limit point to be a $\vartheta(I_2)$-cluster point.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

The authors declare no conflict of interest.