Intuitionistic fuzzy dimension of an intuitionistic fuzzy vector space

: In the present paper the notion of intuitionistic fuzzy dimension of an intuitionistic fuzzy vector space has been developed with the help of intuitionistic fuzzy basis.


Introduction
The notion of intuitionistic fuzzy set (IFS) was introduced by Atanassov [1, 2, 3, 4] as a generalization of Zadeh's fuzzy set [22]. There are situations where IFS theory is more appropriate to dealt with [7]. IFS theory have successfully been applied in knowledge engineering, medical diagnosis, decision making, career determination, etc., [11,21,12]. Several researchers have extended various mathematical aspects such as groups, rings, topological spaces, metric spaces, topological groups, topological vector spaces etc. in IFS [6,10,13,16,17,18,19]. The notion of fuzzy vector subspaces has been introduced by Katsaras [14] and a notion of fuzzy bases and fuzzy dimension was studied by Shi et al. [20]. We have introduced a notion of intuitionistic fuzzy vector space and intuitionistic fuzzy basis in [9]. As a continuation of our paper [9], here we introduced the notion of intuitionistic fuzzy dimension of an intuitionistic fuzzy vector space with the help of intuitionistic fuzzy basis and studied some of its basic results.

Preliminaries
Definition 2.1. [1] Let X be a non-empty set. An intuitionistic fuzzy set (IFS for short) of X is defined as an object having the form A = { x, µ A (x), ν A (x) | x ∈ X}, where µ A : X → [0, 1] and ν A : X → [0, 1] denote the degree of membership (namely µ A (x)) and the degree of non-membership (namely ν A (x)) of each element x ∈ X to the set A, respectively, and 0 ≤µ A (x) + ν A (x) ≤ 1 for each x ∈ X. For the sake of simplicity we shall use the symbol A = (µ A , ν A ) for the intuitionistic In this paper, we use the symbols a ∧ b = min{a, b} and a ∨ b = max{a, b}.
(2) For all x 1 , x 2 , . . . , x n in X, we have Definition 2.7. [9] An IFS V = (µ V , ν V ) of a vector space X over the field K is said to be intuitionistic fuzzy vector space over X if We denote the set of all intuitionistic fuzzy vector spaces over a vector space X by IFV S(X).
Remark 2.8. [9] Let X be a vector space.
(2) If V ∈ IFV S(X), then µ V and ν c V are fuzzy vector subspace of X.
[9] Let V be an intuitionistic fuzzy set in a vector space X. Then, the following are equivalent: (1) V is an intuitionistic fuzzy vector space over X.
(3) For all scalars α, β and for all x, y ∈ X, we have Remark 2.10. [9] Our definition of intuitionistic fuzzy vector space is equivalent to the definition of intuitionistic fuzzy subspace of [19] and [8].
[9] Let V ∈ IFV S(X). Then is a subspace of the vector space X, Then there exists nested collection of subspaces of X as Definition 2.20.