Intuitionistic Fuzzy Structures

Khalid Ebadullah

I. INTRODUCTION

The fuzzy theory has emerged as the most active area of research in many branches of science and engineering. Among various developments of the theory of fuzzy sets[35] a progressive development has been made to find the fuzzy analogues of the classical set theory. In fact the fuzzy theory has become an area of active research for the last 50 years. It has a wide range of applications in the field of science and engineering, e.g. application of fuzzy topology in quantum particle physics that arises in string and $e^{(\infty)}$ -theory of El-Naschie[7-11], electronic engineering, chaos control, computer programming, electrical engineering, nonlinear dynamical system, population dynamics and biological engineering etc.

In [35], Zadeh introduced the theory of fuzzy sets. Atanassov[1-2] introduced and studied the concept of intuitionistic fuzzy sets as a generalization of fuzzy sets. Later on the theory of intuitionistic fuzzy sets caught the interest of various mathematicians round the globe and a huge literature in this direction has been produced in forms of books and research papers published in famous journals round the globe. By reviewing the literature, one can reach them easily, (e. g., see, [1-6], [12], [13], [15], [26], [30] and [31] and the references there in.)

Sequences play an important role in various fields of Real Analysis, Complex Analysis, Functional Analysis and Topology. These are very useful tools in demonstrating abstract concepts through constructing examples and counter examples. The topic "Sequence spaces" is very broad in its own sense as one can study from various points of views. In [22] Kostyrko, Salat and Wilczynski introduced and studied the concept of I-Convergence and later on there has been much progress and development in the study in this direction. (e. g., see, [12], [14-19], [32] and [34] and the references there in.)

In many branches of science and engineering we often come across with different type of sequences and certainly there are situations of inexactness where the idea of ordinary convergence does not work. So to deal with such situations we have to introduce new measures and tools which are suitable to the said situation. Here we give the preliminaries.

II. PRELIMINARIES

Now we quote the following definitions which will be needed in the sequel

Definition 2.1. A binary operation $*: [0,1] \times [0,1] \to [0,1]$ is said to be continuous t-norm if it satisfies the following conditions;

(a) * is associative and commutative.
(b) * is continuous.
(c) $\mathrm{a}^* 1 = \mathrm{a}$ for all $\mathrm{a} \in [0,1]$
(d) $\mathrm{a}^* \mathrm{c} \leq \mathrm{b}^* \mathrm{d}$ whenever $\mathrm{a} \leq \mathrm{b}$ and $\mathrm{c} \leq \mathrm{d}$ for each $\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d} \in [0,1]$ .

For example, $a * b = a \cdot b$ is a continuous t-norm.

Definition 2.2. A binary operation $\diamond: [0,1] \times [0,1] \to [0,1]$ is said to be continuous t-conorm if it satisfies the following conditions;

(a) $\diamond$ is associative and commutative.
(b) $\diamond$ is continuous.
(c) $\mathrm{a} \diamond 0 = \mathrm{a}$ for all $\mathrm{a} \in [0,1]$
(d) $\mathrm{a} \diamondsuit \mathrm{c} \leq \mathrm{b} \diamondsuit \mathrm{d}$ whenever $\mathrm{a} \leq \mathrm{b}$ and $\mathrm{c} \leq \mathrm{d}$ for each $\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d} \in [0,1]$ .

For example, $\mathrm{a} \diamond \mathrm{b} = \min \{a + b, 1\}$ is a continuous t-conorm.

Definition 2.3. Let $^*$ be a continuous t-norm and $\diamond$ be a continuous t-conorm and X be a linear space over the field $(R$ or $C)$ . If $\mu$ and $\nu$ are fuzzy sets on $\mathrm{X}\times (0,\infty)$ satisfying the following conditions, the five- tuple $(\mathrm{X},\mu,\nu,^{*},\diamond)$ is said to be an intuitionistic fuzzy normed space(IFNS) and $(\mu,\nu)$ is called an intuitionistic fuzzy norm. For every x, y $\in \mathbf{X}$ and s, $\mathfrak{t} > 0$

(a) $\mu (\mathrm{x},\mathrm{t}) + \nu (\mathrm{x},\mathrm{t})\leq 1,$ - (b) $\mu (\mathrm{x},\mathrm{t}) > 0$ - (c) $\mu (\mathrm{x},\mathrm{t}) = 1$ iff $\mathrm{x} = 0$ - (d) $\mu (\mathrm{ax},\mathrm{t}) = \mu (\mathrm{x},\frac{t}{|a|})$ for each $a\neq 0$ - (e) $\mu (\mathrm{x},\mathrm{t})*\mu (\mathrm{y},\mathrm{s})\leq \mu (\mathrm{x} + \mathrm{y},\mathrm{t} + \mathrm{s})$ - $(\mathrm{f})\mu (\mathrm{x},\cdot):(0,\infty)\to [0,1]$ is continuous, - (g) $\lim_{t\to \infty}\mu (x,t) = 1$ and $\lim_{t\to 0}\mu (x,t) = 0,$ - $(\mathrm{h})\nu (\mathrm{x},\mathrm{t}) < 1,$
(i) \nu ( $\mathrm{x}$ , $\mathrm{t}$ ) = 0 $\text{iff}$ $\mathrm{x}$ = 0 \quad - (j) \nu ( $\mathrm{a}$ x, $\mathrm{t}$ ) = \nu ( $\mathrm{x}$ ,\frac{\mathrm{t}}{| $\mathrm{a}$ |}) $\text{foreach}$ $\mathrm{a}$ \neq 0 \quad - (k) \nu ( $\mathrm{x}$ , $\mathrm{t}$ ) \diamond \nu ( $\mathrm{y}$ , $\mathrm{s}$ ) \geq \nu ( $\mathrm{x}$ + $\mathrm{y}$ , $\mathrm{t}$ + $\mathrm{s}$ ) \quad - (1) \nu ( $\mathrm{x}$ ,\cdot): (0,\infty) \to [0,1] $\text{iscontinuous,}$ \lim_{t\to \infty} \nu ( $\mathrm{x}$ , $\mathrm{t}$ ) = 1 $\text{and}$ \lim_{t\to 0} \nu ( $\mathrm{x}$ , $\mathrm{t}$ ) = 0, \quad - (n) $\mathrm{a}$ \diamond $\mathrm{a}$ = $\mathrm{a}$ $\text{forall}$ $\mathrm{a}$ \in [0,1].

Definition 2.4. Let $(\mathrm{X},\mu,\nu,^{*},\diamond)$ be IFNS and $(x_{n})$ be a sequence in X. Sequence $(x_{n})$ is said to be convergent to $\mathrm{L}\in \mathrm{X}$ with respect to the intuitionistic fuzzy norm $(\mu,\nu)$ if for every $\epsilon >0$ and $t > 0$ , there exists a positive integer $n_0$ such that $\mu (x_{n} - \mathrm{L},\mathrm{t}) > 1 - \epsilon$ and $\nu (x_{n} - \mathrm{L},\mathrm{t}) < \epsilon$ whenever $n > n_0$ . In this case we write $(\mu,\nu)$ - $\lim x_{n} = \mathrm{L}$ as $n\to \infty$ .

Definition 2.5. If $\mathrm{X}$ be a non-empty set, then a family of set $I \subset P(X)(P(X))$ denoting the power set of $\mathrm{X}$ is called an ideal in $\mathrm{X}$ if and only if

(a) $\phi \in I$
(b) For each $A, B \in I$ , we have $A \cup B \in I$ ;
(c) For each $A\in I$ and $B\subset A$ we have $B\in I$

Definition 2.6. If $X$ be a non-empty set. A non-empty family of sets

$F \subset P(X)(P(X)$ denoting the power set of X) is called a filter on X if and only if

(a) $\phi \notin F$
(b) For each $A, B \in F$ , we have $A \cap B \in F$ ;
(c) For each $A \in F$ and $A \subset B$ we have $B \in F$ .

Definition 2.7. Let $I \subset P(N)$ be a non-trivial ideal and $(\mathrm{X}, \mu, \nu, *, \diamond)$ be an IFNS. A sequence $x = (x_{n})$ of elements in $\mathrm{X}$ is said to be I- convergent to $\mathrm{L} \in \mathrm{X}$ with respect to the intuitionistic fuzzy norm $(\mu, \nu)$ if for every $\epsilon > 0$ and $t > 0$ , The set

In this case $\mathrm{L}$ is called the I-limit of the sequence $(x_{n})$ with respect to the intuitionistic fuzzy norm $(\mu, \nu)$ and we write $I_{(\mu, \nu)} - \lim x_{n} = \mathrm{L}$ .

Definition 2.8. A convergence field of I-convergence is a set

The convergence field $F(I)$ is a closed linear subspace of $\ell_{\infty}$ with respect to the supremum norm, $F(I) = \ell_{\infty} \cap c^{I}$ .

As an insight, while working in the direction of fuzzy theory and Ideal Convergence the author has the following observations

Let N, R and C be the sets of all natural, real and complex numbers respectively. We write

the space of all real or complex sequences.

Listed below are few complementary structures which are intuitionistic fuzzy in nature.

Recently Kostyrko, Šalát and Wilczyński[22], Šalát Tripathy and Ziman[32] introduced and studied the following sequence spaces

Ruckle [27-29] used the idea of a modulus function $f$ to construct the sequence space

Khan and Ebadullah[16] introduced the following sequence spaces

Lindenstrauss and Tzafriri[23] used the idea of Orlicz functions to construct the sequence space

Tripathy and Hazarika[34] introduced the following sequence spaces

Kolk[20-21] gave an extension of $X(f)$ by considering a sequence of modulii $F = (f_k)$ and defined the sequence space

Khan, Suantai and Ebadullah[17] introduced the following sequence spaces

The $\sigma$ -convergent sequences

where $m\geq 0,k > 0$

Khan and Ebadullah[19] introduced the following sequence spaces

Mursaleen [24-25] defined the sequence space $BV_{\sigma}$ , the space of all sequences of $\sigma$ -bounded variation

Khan, Ebadullah and Suantai [18] introduced the following sequence space

Sengönül[33] introduced the Zweier sequence spaces $\mathcal{Z}_0$ and $\mathcal{Z}$ as follows

Khan, Ebadullah and Yasmeen [14] introduced the following classes of sequence spaces

III. MAIN RESULTS

The approach is to construct and study new intuitionistic fuzzy I-convergent sequence spaces

using the $(\mu, \nu)$ intuitionistic fuzzy norm.

IV. CONCLUSION

We can study the algebraic, topological and elementary properties of intuitionistic fuzzy I-convergent sequence spaces $c_{(\mu,\nu)}^{I}, c_{0(\mu,\nu)}^{I}, \mathrm{V}_{\sigma(\mu,\nu)}^{I}, \mathrm{V}_{0\sigma(\mu,\nu)}^{I}, \mathrm{BV}_{\sigma(\mu,\nu)}^{I}, \mathrm{Z}_{(\mu,\nu)}^{I}, \mathrm{Z}_{0(\mu,\nu)}^{I}$ .

Further, as a future research direction, the sequence spaces $c_{(\mu,\nu)}^I$ , $\mathrm{c}_{0(\mu,\nu)}^I$ , $\mathrm{V}_{\sigma(\mu,\nu)}^I$ , $\mathrm{V}_{0\sigma(\mu,\nu)}^I$ , $\mathrm{BV}_{\sigma(\mu,\nu)}^I$ , $\mathrm{Z}_{(\mu,\nu)}^I$ , $\mathrm{Z}_{0(\mu,\nu)}^I$ . can be studied using the Lacunary, Modulus function, Orlicz function, Sequence of moduli, Musielak-Orlicz function and Fibonacci sequences.

The proposed new intuitionistic fuzzy I-convergent sequence spaces are quite pathological from algebraic and topological point of view and one can study it in the direction of double sequences also.

Conflict of Interests: The author declares that there is no conflict of interests.

Research Grant: This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors