‎Intuitionistic Fuzzy Modular Spaces

Transactions on Fuzzy Sets and Systems

مقاله 7، دوره 2، شماره 2، بهمن 2023، صفحه 113-126   

نوع مقاله: Original Article

شناسه دیجیتال (DOI): 10.30495/tfss.2023.1972768.1055

نویسنده

Tayebe Lal Shateri^*

Department of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.

چکیده

‎After the introduction of the concept of fuzzy sets‎ ‎by Zadeh‎, ‎several researches were conducted on‎ ‎the generalizations of the notion of fuzzy sets‎. ‎There are many viewpoints on the notion of metric space in fuzzy topology‎. ‎One of the most important problems in fuzzy topology is obtaining an appropriate concept of fuzzy metric space‎. ‎This problem has been investigated by many authors from different points of view‎. ‎Atanassov gives the concept of the intuitionistic fuzzy set as a generalization of the fuzzy set‎. ‎Park introduced the notion of intuitionistic fuzzy metric space as a natural generalization of fuzzy metric spaces due to George and Veeramani‎. ‎This paper introduces the concept of intuitionistic fuzzy modular space‎. ‎Afterward‎, ‎a Hausdorff topology induced by a $\delta$-homogeneous intuitionistic fuzzy modular is defined and some related topological properties are also examined‎. ‎After giving the fundamental definitions and the necessary examples‎, ‎we introduce the definitions of intuitionistic fuzzy boundedness‎, ‎intuitionistic fuzzy compactness‎, ‎and intuitionistic fuzzy convergence‎, ‎and obtain several preservation properties and some characterizations concerning them‎. ‎Also‎, ‎we investigate the relationship between an intuitionistic fuzzy modular and an intuitionistic fuzzy metric‎. ‎Finally‎, ‎we prove some known results of metric spaces including Baire’s theorem and the Uniform limit theorem for intuitionistic fuzzy modular spaces‎.

کلیدواژه‌ها

Fuzzy set؛ Modular space؛ Fuzzy modular space؛ Intuitionistic fuzzy modular space‎

مراجع

1] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst, 20 (1986), 87-96.
[2] D. Coker, An introduction to intuitionistic fuzzy topological spaces, Fuzzy Sets Syst, 88 (1997), 81-89.
[3] M. S. El Naschie, On the uncertainty of Cantorian geometry and two-slit experiment, Chaos, Soliton & Fractals, 9(3) (1998), 517-529.
[4] M. S. El Naschie, On the verifications of heterotic strings theory and ϵ(∞)theory, Chaos, Soliton & Fractals, 11(2) (2000), 2397-2407.
[5] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets Syst, 27 (1988), 385-389.
[6] A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets Syst, 64 (1994), 395-399.
[7] S. J. Kilmer, W. M. Kozlowski and G. Lewicki, Best approximants in modular function spaces, J. Approximation Theory, 63(3) (1990), 338-367.
[8] W. M. Kozlowski, Notes on modular function spaces I, Comment. Math, 28(1) (1988), 87-100.
[9] W. M. Kozlowski, Notes on modular function spaces II, Comment. Math, 28(1) (1988), 101-116.
[10] W. M. Kozlowski and G. Lewicki, Analyticity and polynomial approximation in modular function spaces, J. Approximation Theory, 58(1) (1989), 15-35.
[11] I. Kramosil and J. Mich´ alek, Fuzzy metric and statistical metric spaces, Kybernetika, 11(5) (1975), 336-344.
[12] W. A. Luxemburg, Banach function spaces, Ph.D. thesis, Delft University of Technology, Delft, The Netherlands, (1959).
[13] J. Musielak and W. Orlicz, On modular Spaces, Studia Math, 18 (1959), 49-56.
[14] H. Nakano, Modulared Semi-Ordered Linear Spaces, Tokyo Math. Book Ser, 1, Maruzen Co. Tokyo, (1950).
[15] K. Nourouzi, Probabilistic modular spaces, Proceedings of the 6th International ISAAC Congress, Ankara, Turkey, (2007).
[16] K. Nourouzi, Baire’s theorem in probabilistic modular spaces, Proceedings of the World Congress on Engineering (WCE 08), 2 (2008), 916-917.
[17] J. H. Park, Intuitionistic fuzzy metric spaces, Chaos Solitons Fractals, 22 (2004), 1039-1046.
[18] B. Schweizer and A. Sklar, Statistical metric spaces, Pac. J. Math, 10 (1960), 314-334.
[19] Y. Shen and W. Chen, On fuzzy modular spaces, J. Appl. Math, Article ID 576237, 2013 (2013), 1-8.
[20] L. A. Zadeh, Fuzzy sets, Inf. Cont, 8 (1965), 338-353.