The notation of fuzzy set field is introduced. A fuzzy metric is redefined on fuzzy set field and on arbitrary fuzzy set in a field. The metric redefined is between fuzzy points and constitutes both fuzziness and crisp property of vector. In addition, a fuzzy magnitude of a fuzzy point in a field is defined. 1. Introduction Different researchers introduced the concept of fuzzy field and notion of fuzzy metric on fuzzy sets. How to define a fuzzy metric on a fuzzy set is still active research topic in fuzzy set theory which is very applicable in fuzzy optimization and pattern recognition. The notion of fuzzy sets has been applied in recent years for studying sequence spaces by Tripathy and Baruah [1], Tripathy and Sarma [2], Tripathy and Borgohain [3], and others. Wenxiang and Tu [4] introduced the concept of fuzzy field in field and fuzzy linear spaces over fuzzy field. Furthermore, different authors are attempting to define fuzzy normed linear spaces, fuzzy inner product space, fuzzy Hilbert space, fuzzy Banach spaces, and so forth (cf. [5–8]). Many authors introduced different notion of fuzzy metric on a fuzzy set from different points of view. Kaleva and Seikkala [9] introduced the notion of a fuzzy metric space where metric was defined between fuzzy sets. The idea behind this notion was to fuzzify the classical metric by replacing real values of a metric by fuzzy values (fuzzy numbers). For the further research work and the properties of this type of fuzzy metric space see for instance Fang [10], Quan Xia and Guo [11], and others. Wong [12] defined fuzzy point and discussed its topological properties and there after Deng [13] defined Pseudo-metric spaces where metric was defined between fuzzy points rather than between fuzzy sets. Hsu [14] introduced fuzzy metric space with metric defined between fuzzy points and examined the completion of fuzzy metric space. For different notions of fuzzy metric space and for further research work see for instance Shi [15], Shi and Zheng [16], Shi [17] and others. This paper is an attempt to define a fuzzy set field in a field which is assumed to be the generalization of a fuzzy field introduced by [4]. We restate fuzzy set in more general form by allowing a particular fuzzy set to consist a family of membership functions. A fuzzy metric on fuzzy set and on fuzzy set field is reintroduced in such way that the classical metric is considered as a special type of fuzzy metric. In the sequel, a notion of magnitude of a fuzzy point in a field is introduced for the first time (up to our knowledge) and some of its
References
[1]
B. C. Tripathy and A. Baruah, “N？rlund and Riesz mean of sequences of fuzzy real numbers,” Applied Mathematics Letters, vol. 23, no. 5, pp. 651–655, 2010.
[2]
B. C. Tripathy and B. Sarma, “Some double sequence spaces of fuzzy numbers defined by Orlicz functions,” Acta Mathematica Scientia, vol. 31, no. 1, pp. 134–140, 2011.
[3]
B. C. Tripathy and S. Borgohain, “Sequence spaces of fuzzy real numbers using fuzzy metric,” Kyungpook Mathematical Journal, vol. 54, no. 1, pp. 11–22, 2014.
[4]
G. Wenxiang and L. Tu, “Fuzzy linear spaces,” Fuzzy Sets and Systems, vol. 49, no. 3, pp. 377–380, 1992.
[5]
A. Taghavi and M. Mehdizadh, “Adjoint operator in fuzzy normed linear spaces,” The Journal of Mathematics and Computer Science, vol. 2, pp. 453–458, 2011.
[6]
G. Seob Rhe and I. Hwang, “On fuzzy complete normed spaces,” Journal of the Changcheong Mathematical Society, vol. 22, no. 2, 2009.
[7]
R. Saadati and S. M. Vaezpour, “Some results on fuzzy banach spaces,” Journal of Applied Mathematics and Computing, vol. 17, no. 1-2, pp. 475–484, 2005.
[8]
C. P. Santhosh and T. V. Ramakrishnan, “Norm and inner product on fuzzy linear spaces over fuzzy fields,” Iranian Journal of Fuzzy Systems, vol. 8, no. 1, pp. 135–144, 2011.
[9]
O. Kaleva and S. Seikkala, “On fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 12, no. 3, pp. 215–229, 1984.
[10]
J. Fang, “An improved completion theorem of fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 166, pp. 65–74, 2011.
[11]
Z. Quan Xia and F. F. Guo, “Fuzzy metric spaces,” Applied Mathematics and Computation, vol. 16, pp. 371–381, 2004.
[12]
C. K. Wong, “Fuzzy points and local properties of fuzzy topology,” Journal of Mathematical Analysis and Applications, vol. 46, pp. 316–328, 1974.
[13]
Z. Deng, “Fuzzy pseudo-metric spaces,” Journal of Mathematical Analysis and Applications, vol. 86, no. 1, pp. 74–95, 1982.
[14]
N. Hsu, “On completion of fuzzy metric spaces,” Journal of National Taiwan Normal University, vol. 37, pp. 385–392, 1992.
[15]
F.-G. Shi, “ -fuzzy metric spaces,” Indian Journal of Mathematics, vol. 52, no. 2, pp. 231–250, 2010.
[16]
F. Shi and C. Zheng, “Metrization theorems in L-topological spaces,” Fuzzy Sets and Systems, vol. 149, no. 3, pp. 455–471, 2005.
[17]
F.-G. Shi, “Pointwise pseudo-metrics in -fuzzy set theory,” Fuzzy Sets and Systems, vol. 121, no. 2, pp. 209–216, 2001.
[18]
P. M. Pu and Y. M. Liu, “Fuzzy topology. I. Neighborhood structure of a fuzzy point and Moore-Smith convergence,” Journal of Mathematical Analysis and Applications, vol. 76, no. 2, pp. 571–599, 1980.
[19]
H. Nasseri, “Fuzzy numbers: positive and nonnegative,” International Mathematical Forum, vol. 3, no. 36, pp. 1777–1780, 2008.
[20]
H. J. Zimmermann, Fuzzy Set Theory and Its Applications, Kluwer Academic Publishers, Boston, Mass, USA, Second edition, 1996.
[21]
D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academic Press, San Diego, Calif, USA, 1980.