Ideal Statistically Pre-Cauchy Triple Sequences of Fuzzy Number and Orlicz Functions

In this paper, we extend the notions of ideal statistically convergence for sequence of fuzzy number. We introduce the notions ideal statistically pre-Cauchy triple sequences of fuzzy number about Orlicz function, and give some correlation theorem. It is shown that ideal statistically pre-Cauchy if and only if ≤ ≥ ∈ . At the same time, we have proved { } ijk x x = is ideal statistically 0 x if and   ∈ × × ≥ ∈     


Introduction
The notion of statistical convergence was introduced by Fast [1] and also independently by Buck [2] and Schoenberg [3] for real and complex sequences.Over the years and under different names statistical convergence has been discussed in the theory of Fourier analysis, Ergodic theory and Number theory. Later on it was further investigated from the sequence spaces point of view and linked with summability theory by Altinok and Et [4], Connor [5], Et et al. ([6] [7] [8]), Fridy [9], Fridy and Orhan [10], Mursaleen [11] and many others. Matloka [12] defined the notion of fuzzy sequence and introduced bounded and convergent sequences of fuzzy real numbers and studied their some properties. After then, Nuray and Savas [13] defined the notion of statistical convergence for sequences of fuzzy numbers. Since then, there has been increasing interest in the study of statistical convergence of fuzzy sequences (see [14]- [19]). Lindesstrauss and Tzafriri [20] used the idea of Orlicz sequence space, 1 : : , which is Banachi space with the norm: The space M l is closely related to the space p l , which is an Orlicz sequence space with Connor, Fridy and Kline [21] proved that statistical convergent sequences are statistically pre-Cauchy and any bounded statistically pre-Cauchy sequence with nowhere dense set of limit points is statistically convergent. They also gave an example showing statistically pre-Cauchy sequences are not necessarily statistically convergent.
In this paper, we extend the notions of ideal statistically convergence for sequence of fuzzy number. We introduce the notions ideal statistically pre-Cauchy triple sequences of fuzzy number about Orlicz function, and give some correlation theorem. Also, some properties of these new sequence spaces are investigated. It popularized the work of predecessors.

Definitions and Preliminaries
In this section, we give some basic notions which will be used throughout the paper.

( )
Using the results of [22] [23], we see that  The concept of Orlicz function was introduced by Parashar and Choudhary [25], A mapping

4) M is continuous, nondecreasing and convex.
An Orlicz function may be bounded or unbounded. For example, A triple sequence can be defined as a function  where the vertical bars indicate the number of elements in the set [28].

Main Results
Definition 3.1. A triple sequence of fuzzy numbers is said to be ideal statistically pre-Cauchy if for every 0, 0 where the I denote the nontrivival ideal of N.
Now suppose that x is ideal statistically pre-Cauchy and that ε has been given. Let Since M is bounded Orlicz function, there exist an integer G such that , , : : .
We have x is ideal statistically convergent to 0 x . Now suppose that x is ideal statistically convergent to 0 x , let 0, 0 Since M is bounded Orlicz function, there exist an integer G such that Note that, for each , , m n t N ∈ ( )

Conclusion
In this article, we introduced ideal statistically pre-Cauchy triple sequences of fuzzy numbers about Orlicz function. At the same time, we have proved some properties and relationships.

Fund
This work is supported by National Natural Science Fund of China (11761056); the Natural Science Foundation of Qinghai Province (2020-ZJ-920); University level planning project of Qinghai Minzu University (2021XJGH24).

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.