454 International Journal of Fuzzy Systems, Vol. 14, No. 3, September 2012 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 : 0,1 where is a nonempty set. Then several researches were conducted on the generalizations of the notion of fuzzy set. The idea of intuitionistic 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,, : where the functions : 0,1 and : 0,1 define respectively, the degree of membership and degree of non-membership of the element to the set, which is a subset of, and for every, 1. 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] considered the concept of intuitionistic fuzzy sets with the group theory. Akram [1] introduced Lie algebras into the intuitionistic fuzzy sets. Then, Chen [5] studied the intuitionistic fuzzy Lie sub-superalgebras. Park [14] defined 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 intuitionistic 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 Edelstein 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 induced 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 intuitionistic 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 extensions 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 intuitionistic 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 intuitionistic 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. 2012 TFSA
Banu Pazar Varol and Halis Aygün: Intuitionistic Fuzzy Metric Groups 455 Definition 2.1 [17]: A binary operation : 0,1x0,1 0,1 is a continuous t-norm if satisfies the following conditions: (1),,0,1 (2),,,0,1 (3) is continuous (4) 1, 0,1 (5) whenever and,,,, 0,1. Definition 2.2 [17]: A binary operation : 0,1x0,1 0,1 is a continuous t-conorm if satisfies the following conditions: (1),,0,1 (2),,,0,1 (3) is continuous (4) 0, 0,1 (5) whenever and,,,, 0,1 Note 2.3: The concepts of t-norms (triangular norms) and t-conorms (triangular conorms) are known as the axiomatic skeletons that we use for characterizing fuzzy intersections and unions, respectively. Note 2.4: (1) For any, 0,1 with, there exist, 0,1 such that and. (2) For any 0,1, there exist, 0,1 such that and. Definition 2.5 [8, 9]: A fuzzy metric space is a triple,, such that is a nonempty set, is a continuous t-norm and is a fuzzy set on xx0, satisfying the following conditions, for all,,,, 0: (1),, 0; (2),, 1 if and only if ; (3),,,,; (4),,,,,,; (5),,. : 0, 0,1 is continuous. If,, is a fuzzy metric space, we say that, is a fuzzy metric on. Example 2.6 [8]: Let, be a metric space. Define,,0,1 and let be a function on x0, defined by,,, Then,, is a fuzzy metric space and, is called the fuzzy metric induced by. Definition 2.7 [14]: An intuitionistic fuzzy metric spaces is a 5-tuple,,,, such that is a nonempty set, is a continuous t-norm, is a continuous t-conorm and, are fuzzy sets on xx0, satisfying the following conditions, for all,,,, 0: (1),,,,1; (2),, 0; (3),, 1 if and only if ; (4),,,,; (5),,,,,,; (6),,. : 0, 0,1 is continuous; (7),, 0; (8),, 0 if and only if ; (9),,,,; (10),,,,,,; (11),,. : 0, 0,1 is continuous. If,,,, is an intuitionistic fuzzy metric space, we say that,,, (or simply, ) is an intuitionistic fuzzy metric on. The functions,, and,, denote the degree of nearness and the degree of non-nearness between and with respect to t, respectively. Remark 2.8 [12]: If,,,, is an intuitionistic fuzzy metric space, then,, is a fuzzy metric space in the sense of Definition 2.5. Conversely, if,, is a fuzzy metric space, then,,1,, is an intuitionistic fuzzy metric space, where 11 1 for any, 0,1. Remark 2.9: In intuitionistic fuzzy metric spaces X,,,. is non-decreasing and,,. is non-increasing for all,. Example 2.10 [14] : Let, be a metric space. Denote and min1, for all, 0,1 and let and be fuzzy sets on xx0, defined by,,,,,,, for all,,,. Then,,,, is an intuitionistic fuzzy metric space. Remark 2.11 [14] : The Example 2.10 holds even with t-norms min, and the t-conorms max,. Hence,, is an intuitionistic fuzzy metric with respect to any continuous t-norms and continuous t-conorms. Let take 1 in Example 2.10, then we have,,,,,,, This intuitionistic fuzzy metric is called induced intuitionistic fuzzy metric by a metric. Example 2.12 [14]: Let. Define max 0,1 and for all, 0,1 and let and be fuzzy sets on xx0, as follows:
456 International Journal of Fuzzy Systems, Vol. 14, No. 3, September 2012,,,,,,,, for all, and 0. Then,,,, is an intuitionistic fuzzy metric space. Remark 2.13: In the above example, t-norm and t-conorm are not associated. There exist no metric on satisfying,,,,,,,, where,, and,, are as defined in above example. The above functions, is not an intuitionistic fuzzy metric with t-norm and t-conorm defined as min, and max,. Definition 2.14 [14]: Let,,,, be an intuitionistic fuzzy metric space. Given, 0,1 and 0. The set,, :,, 1,,, is called the open ball with center and radius with respect to. Theorem 2.15 [14]: Every open ball,, is an open set. The collection,, :, 0,1, 0, where,, :,, 1,,, for all, 0,1 and 0, is a base of a topology on and this topology is denoted by,. Hence, each intuitionistic fuzzy metric, on generates a topology, on. Notice that the collection,, :, is also a base for,. The topology, is Hausdorff and first countable. If, is a metric space, then the topology induced by coincides with the topology, induced by the intuitionistic fuzzy metric,. The family :, where, x,, 1,,, is a base for uniformity, on whose induced topology coincides with the topology, and hence,, is metrizable. Theorem 2.16 [14]: Every intuitionistic fuzzy metric spaces is Hausdorff. Remark 2.17 [14]: Let, be a metric space. Let,,,,,,,,, be the intuitionistic fuzzy metric defined on. Then the topology induced by the metric and the topology, induced by the intuitionistic fuzzy metric, are the same. Definition 2.18 [14]: Let,,,, be an intuitionistic fuzzy metric space. (1) A sequence on is called Cauchy sequence if for each 0 and 0 there exist such that,, 1 and,, for all,. (2),,,, is called complete intuitionistic fuzzy metric space if every Cauchy sequence is convergent with respect to,. If,,,, is a complete intuitionistic fuzzy metric space we say that, is a complete intuitionistic fuzzy metric on. If is a Cauchy sequence on,,, then is a Cauchy sequence on the intuitionistic fuzzy metric space,,,,, so the uniform space,, is complete whenever,,,, is a complete. Definition 2.19 [15]: A fuzzy metric group is a 4-tuple,,, such that,, is a fuzzy metric space and,, is a topological group. 3. Intuitionistic Fuzzy Metric Groups Definition 3.1: A topological space, is called intuitionistic fuzzy metrizable if there exist an intuitionistic fuzzy metric, on such that the topology, induced by, coincides with. The intuitionistic fuzzy metric is called compatible with. A topological group is called intuitionistic fuzzy metrizable if it is intuitionistic fuzzy metrizable as a topological spaces. Definition 3.2: An intuitionistic fuzzy metric group is a 6-tuple,,,,, such that,,,, is an intuitionistic fuzzy metric space and,,, is a topological group. Definition 3.3: An intuitionistic fuzzy metric, on a group is called left invariant (right invariant) if,,,, and,,,, (,,,, and,,,, ) whenever,, and 0. Theorem 3.4: If is an intuitionistic metrizable topological group and, is a compatible left (right) invariant intuitonistic fuzzy metric on, then the uniform structure,, induced by, coincides with the left (right) uniform structure,.
Banu Pazar Varol and Halis Aygün: Intuitionistic Fuzzy Metric Groups 457 Proof: We must show that the two uniformities have a common base. Let, for each, take a basic neighborhood of,, :,, 1,,, Hence, for, :, :,, 1,,,, :,, 1,,, Since, is left invariant,,. In the following, we use next well-known theorem. Theorem 3.5: Let, be a group with identity and let be a family of subsets of which satisfies the following properties: (1) (2) If,, then there exists such that (3) for every, there exist such that (4) for every and every, there exist such that. Then, there exist a unique Hausdorff topological group on which has as a local base of. Lemma 3.6: Let, be an group and, be a left invariant intuitionistic fuzzy metric on. Then,, is symmetric for all and,,.,, for all. Proof: Let,,. Then,, 1 and,,,, 1 and,,,, 1and,,,,. From the following equalities we see that the second part of the lemma., 1, 1 :,, 1 1 1 and,, 1 1 :,, 1 1 1 and,, 1 1 :, 1, 1 :., 1, 1., 1, 1 Proposition 3.7: A topological group approves a compatible left invariant intuitionistic fuzzy metric if and only if it approves a compatible right invariant intuitionistic fuzzy metric. Proof: Let, be a compatible left invariant intuitionistic fuzzy metric on a topological group. Define two functions and on xx0, as follows:,,,, x, y, t,, whenever, and 0., is a right invariant intuitionistic fuzzy metric on. Indeed,,,,,,,,,,, Similarly,,, x, y, t. Now, we will show that it is a compatible one. Since,, is symmetric for all, then we have,,,,. Hence,,, is a base of neighbourhoods of and that every topological group is homogeneous. From now on the paper all topologies are assumed to be. Theorem 3.8: Let, be a group and, be a left and right invariant intuitionistic fuzzy metric on. Then,,,,, is an intuitionistic fuzzy metric group. Proof: Let,, : be the neighbourhood base of for the intuitionistic fuzzy topology,. We need to show that satisfies conditions (1)-(4) in the Theorem 3.5. Since, is a Hausdorff topology and is a neighbourhood base of, we only check the conditions (3)-(4). (3) Let,,. From the Lemma 3.6,,, is symmetric. So, if we find such that,,.,,,,, then it is sufficient. Let such that for each, 1 1 1, and let max,2. Then, for each,,,,
458 International Journal of Fuzzy Systems, Vol. 14, No. 3, September 2012,.,,.,,,,.,,, 1,, 1 1 1 1 and,.,,.,,,,.,,, 1,, 1 So,.,,. (4) Let,,, and.,,. Then,.,,,,. Hence,,,,, 1 and,,,,. So,,,. This completes the proof. 4. Intuitionistic Fuzzy Metric On Quotient Groups Let be a complete topological group and be a closed normal subgroup of. In general, the qutient group is not complete. But if is metrizable, the quotient group is complete. Moreover, if and are complete, then the group is complete. [15] Let, be a metric group and be a closed normal subgroup of. Then the metric induced by on the quotient group is defined by, inf, :,. The fuzzy metric induced by on is defined by following:,, sup,, :,, 0 The induced fuzzy metric, on for a closed normal subgroup of a fuzzy metric group,,, is defined by,, sup,, :,, 0. [15] Now we consider these results for intuitionistic fuzzy metric groups. The intuitionistic fuzzy metric, induced by on is defined by the following:,, sup,, :,,, inf,, :,, 0. Let be an intuitionistic fuzzy metrizable topological group and, be a left invariant compatible intuitionistic fuzzy metric on. Let be a closed normal subgroup of. The induced intuitionistic fuzzy metric x0, is defined by, on x,, sup,, :,,, inf,, :,, 0 Theorem 4.1:, satisfies the conditions (2)-(5) and (7)-(10) which is given in the definition of intuitonistic fuzzy metric. Proof: Conditions (2), (4), (7) and (9) are obvious. If and 0, then,, 1 and,, 0 for all 0. Conversely, let cosets,,,, 1 and,, 0 for all 0. Then there exist two sequences, satisfying,, 1 and,, for all. We will show that converges to. Let take a neighbourhood,, of the identity. Consider. Then,,,,,,, 1 1 and,,,,,, Therefore, for each,,, and thus, the sequence converges to. Since is normal. So, for each. Since is closed, then. Thus,. Similarly, we can see that the sequence converges to and. Thus (3) and (8) are provided. Let,. Since and are continuous, we have,,,,,,
Banu Pazar Varol and Halis Aygün: Intuitionistic Fuzzy Metric Groups 459,, and,,,,,,,, for each,. Thus (5) and (9) are provided. In general, and do not satisfy conditions (6) and (11) of the definition of an intuitionistic fuzzy metric. Example 4.2: Let (,+) be the usual group of real numbers and let consider the continuous t-norm and the continuous t-conorm on the unit interval I=[0,1] defined by and min 1,. Let and be two sequences of continuous non-decreasing and non-increasing functions from 0, into, satisfying that the functions and are not continuous, respectively. Let and be two functions from 0, into [0,1] defined by,,, max, 1 1,,,, min, 0, where [x] means for the integer part of the real numbers x. Claim 1., is an intuitionistic fuzzy metric on 0, such that 0,,,, and 0,,,, for each,, 0,. Proof: We only need to check the conditions,,,,,, and,,,,,, where,, and, 0. Case 1. If,,,, 1 and,,,, 0, then and consequently,, 1 and,, 0. Case 2. Let,, 1,,, and,, 0,,, for some. Since, then,,. Because of is non-decreasing and is non-increasing the result is provided. Case 3. If,, 1,,, and,, 0,,, for some, then the result is similarly to the Case 2. Case 4. If,,,,, and,,,,, for some,, then for every 0 and. for every 0 and. Let and be two functions from x x0, into [0,1] defined by,, sup,, :,,, inf,, :, Claim 2. The functions,, and,, are not continuous on 0,, where denotes the set of integer numbers. Proof: Let and, then,, : Let and, then,, : Consequently,,, and,, are not continuous. From Theorem 4.1 and Example 4.2, we see that definition of intuitionistic fuzzy metric space must be modified by replacing conditions (6) and (11) by 6 the function,, is lower semicontinuous for each, 11 the function,, is upper semicontinuous for each, By a lower-upper intuitionistic fuzzy metric space we mean a 5-tuble,,,, such that is a nonempty set, is a continuous t-norm, is a continuous t-conorm, and are two fuzzy set on xx0, satisfying conditions (1)-(5), (7)-(10) and 6, 11 above. Then,, is called a lower-upper intuitionistic fuzzy metric on. In [14] Park proved that if, is an intuitionistic fuzzy metric, then,, is non-decreasing and,, is non-increasing for all,. Since for any lower-upper intuitionistic fuzzy metric, and any, 0 with, we have,,,,,,,, and,,,,,,,, for all,, we obtain that Park s result remains valid for lower-upper intuitionistic fuzzy metric. Let see the following results: (1) Every lower-upper intuitionistic fuzzy metric, on induces a metrizable topology, on such that,, : is a neighbourhood base of for each.
460 International Journal of Fuzzy Systems, Vol. 14, No. 3, September 2012 (2) If, is a lower-upper intuitionistic fuzzy metric on, then : is a base for a uniformity on compatible with,, where, :,, 1,,, for all. If we introduce the notions of a lower-upper intuitionistic fuzzy metric group and of a lower-upper intuitionistic fuzzy metrizable topological group, it is easily seen that the results which we have obtained above also hold for lower-upper intuitionistic fuzzy metric groups and for lower-upper intuitionistic fuzzy metrizable topological groups, respectively. Let,,,,, be a lower-upper intuitionistic fuzzy metric group and be a closed normal subgroup of. Then we define the set,, by, 1, 1,, 1 11,,, 1 1 Lemma 4.3: Let,,,,, be a lower-upper intuitionistic fuzzy metric group such that, is left invariant and let be a closed normal subgroup of. Then,,,, for each. Proof: Let. If,,, then,,,, 1 and,,,,. So,,,. Conversely, if,,, we can find and such that,, 1 and,,. Since,,,, 1,,,,, and, we obtain that. Theorem 4.4: Let,,,,, be a lower-upper intuitionistic fuzzy metric group such that, is left invariant and is a closed normal subgroup of. Then the topology of the lower-upper intuitionistic fuzzy metric group,,,,, coincides with the quotient topology. Proof: The family,, is a base of neighbourhoods of in for the quotient topology. By Lemma 4.3 every element of,, belongs to the family,, So, we must. prove that every neighbourhood of for the quotient topology is also a neighbourhood of for the intuitionistic fuzzy topology. Let be a set such that is a neighbourhood of in. We can choose,, such that,,. Then,,,. By Lemma 4.3,,, is a neighbourhood of for the intuitionistic fuzzy topology. This completes the proof. 5. Conclusions Intuitionistic fuzzy sets have the forceful ability of explaining uncertainties compared with fuzzy sets. In this paper, we generalize the approach introduced by Romaguera et al. [15]. In this manner, we define the notion of intuitionistic fuzzy metric group and study some properties. Acknowledgment The authors are highly grateful to the referees for their valuable suggestions. References [1] M. Akram, Intuitionistic fuzzy Lie ideals of Lie algebras, Int. Journal of Fuzzy Math., vol. 16, no. 4, pp. 991-1008, 2008. [2] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, vol. 20, pp.87-96, 1986. [3] R. Biswas, Intuitionistic fuzzy subgroups, Math. Forum, vol. 10, pp. 37-46, 1989. [4] C. L. Chang, Fuzzy topological spaces, J. Math. Appl., vol. 24, pp. 182-193, 1968. [5] W. Chen, Intuitionistic fuzzy quotient Lie superalgebras, International Journal of Fuzzy Systems, vol. 12, no. 4, pp. 330-339, 2010. [6] D. Çoker, An introduction to intuitionistic fuzzy topological spaces, Fuzzy Sets and Systems, vol. 88, pp. 81-89, 1997. [7] R. Engelking, General Topology, Warsaw, Polish,Sci.Publ, 1997. [8] A. George and P.Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems, vol. 64, pp. 395-399, 1994. [9] A. George and P. Veeramani, On some results of analysis for fuzzy metric spaces, Fuzzy Sets and Systems, vol. 90, pp. 365-368, 1997. [10] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets and Systems, vol. 27, pp. 385-389, 1988.
Banu Pazar Varol and Halis Aygün: Intuitionistic Fuzzy Metric Groups 461 [11] V. Gregori and S. Romaguera, Some properties of fuzzy metric spaces, Fuzzy Sets and Systems, vol. 115, pp. 485-489, 2000. [12] V. Gregori, S. Romaguera, and P. Veeramani, A note on intuitionistic fuzzy metric spaces, Chaos, Solitons and Fractals, vol. 28, pp. 902-905, 2006. [13] O. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetika, vol. 11, pp. 326-334, 1975. [14] J. H. Park, Intuitionistic fuzzy metric spaces, Chaos, Solitons and Fractals, vol. 22, pp. 1039-1046, 2004. [15] S. Romaguera and M. Sanchis, On fuzzy metric groups, Fuzzy Sets and Systems, vol. 124, pp. 109-115, 2001. [16] R. Saadati and J. H. Park, On the intuitionistic fuzzy topological spaces, Chaos, Solitons and Fractals, vol. 27, pp. 331-344, 2006. [17] B. Schweizer and A. Sklar, Statistical metric space, Pacific J. Maths., vol. 10, pp. 314-334, 1960. [18] L. A. Zadeh, Fuzzy sets, Inform Control, vol. 8, pp. 338-353, 1965.