454 International Journal of Fuzzy Systems, Vol. 14, No. 3, September 2012 © 2012 TFSA Intuitionistic Fuzzy Metric Groups Banu Pazar Varol and Halis Aygün Abstract 1 The aim of this paper is to introduce the structure of the called “intuitionsitic fuzzy metric group” and after giving fundamental definitions, to study some properties of intuitionistic fuzzy metric group. Keywords: Topology, fuzzy metric spaces, intuitionistic fuzzy metric spaces, group, normal subgroup. 1. Introduction Zadeh [18], in 1965, introduced the concept of fuzzy set as a mapping :ߣ ՜ ሾͲ,ͳሿ where  is a nonempty set. Then several researches were conducted on the gener- alizations of the notion of fuzzy set. The idea of in- tuitionistic fuzzy set was initiated by Atanassov [2] as a generalization notion of fuzzy set. By an intuitionistic fuzzy set we mean an object having the form ܣൌ ሼ൏ ߤ ,ݔ ஺ ሺݔሻ, ߴ ஺ ሺݔሻ ൐: אݔሽ where the functions ߤ ஺ :՜ ሾͲ,ͳሿ and ߴ ஺ :  ՜ ሾͲ,ͳሿ define respectively, the degree of membership and degree of non-membership of the element אݔ to the set ܣ, which is a subset of , and for every אݔ, ߤ ஺ ሺݔሻ൅ ߴ ஺ ሺݔሻ൑ͳ. Then, using the concept of intuitionistic fuzzy set several structures have been constructed. Çoker [6] introduced the concept of intuitionistic fuzzy topological spaces. The theory of intuitionistic fuzzy topology was arised from the theory of fuzzy topology in Chang’s sense [4]. Biswas [3] consid- ered the concept of intuitionistic fuzzy sets with the group theory. Akram [1] introduced Lie algebras into the in- tuitionistic fuzzy sets. Then, Chen [5] studied the in- tuitionistic fuzzy Lie sub-superalgebras. Park [14] de- fined the notion of intuitionistic fuzzy metric space with the help of continuous t-norm and continuous t-conorm. Then, Saadati and Park [16] developed the theory of in- tuitionistic fuzzy topological spaces (both in metric and normed). The idea of fuzzy metric spaces based on the notion of continuous t-norm was initiated by Kramosil and Michalek [13]. Then, Grabiec [10] extended the Corresponding Author: B. Pazar Varol is with the Department of Mathematics, Kocaeli University, Umuttpe Campus, 41380, Kocael,Turkey. E-mail: banupazar@kocaeli.edu.tr, halis@kocaeli.edu.tr Manuscript received 17 Feb. 2011; revised 19 Jun. 2012; accepted 27 Aug. 2012. well-known fixed point theorems of Banach and Edel- stein to fuzzy metric spaces. The notion of fuzzy metric space plays a fundamental role in fuzzy topology, so many authors have studied several notions of fuzzy metric. Somewhat reformulating the Kramosil and Michalek’ s definition, George and Veeramani ([8, 9]) introduced and studied fuzzy metric spaces and topological spaces in- duced by fuzzy metric. They showed that every metric induces a fuzzy metric. Then, Gregori and Romaguera [11] proved that the topology generated by a fuzzy metric is metrizable. They also showed that if the fuzzy metric space is complete, the generated topology is completely metrizable. In 2004, Park [14] introduced the concept of intuitionistic fuzzy metric spaces as a generalization of fuzzy metric spaces due to Geroge and Veeramani [8]. Then, Gregori et al. [12] studied on intuitionistic fuzzy metric spaces and showed that for each intuitionistic fuzzy metric space, the topology generated by the in- tuitionistic fuzzy metric coincides with the topology generated by the fuzzy metric. Both of fuzzy metric spaces and intuitionistic fuzzy metric spaces have many applications in various areas of mathematics. One of them is the concept of fuzzy metric group. In 2001, Romaguera and Sanchis [15] defined the concept of fuzzy metric groups which gives us to exten- sions of classical theorems on metric groups to fuzzy metric groups. They also studied some properties of the quotient subgroups of a fuzzy metric group. In the present paper, we consider the notion of in- tuitionistic fuzzy metric spaces with the group theory. The main idea of this work is to extend the fuzzy metric group to intuitonistic fuzzy metric setting. The structure of this paper is as follows: In Section 2, as a preliminaries, we give some definitions and uniform structures of fuzzy metric spaces and intuitonistic fuzzy metric spaces. In Section 3, we first introduce the notion of intuitionistic fuzzy metric group and then study several results of in- tuitionistic fuzzy metric groups as a generalization of fuzzy metric groups. In last section, we analyze properties of the quotient subgroups of an intuitionistic fuzzy metric group. In this study our basic reference for topology is Engelking’s book [7]. 2. Preliminaries Throughout this paper let  be a nonempty set.