Document Type : Research Paper


1 Biochemical engineering college, Beijing Union University, Beijing 100023, P. R. China

2 Department of Mathematics, School of Science, Beijing Institute of Technology, Beijing 100081, P. R. China


In this paper, rstly, it is proved that, for a fuzzy vector space,
the set of its fuzzy bases de ned by Shi and Huang, is equivalent to the family
of its bases de ned by P. Lubczonok. Secondly, for two fuzzy vector spaces,
it is proved that they are isomorphic if and only if they have the same fuzzy
dimension, and if their fuzzy dimensions are equal, then their dimensions are
the same, however, the converse is not true. Finally, fuzzy dimension of direct
sum is considered, for a nite number of fuzzy vector spaces and it is proved
that fuzzy dimension of their direct sum is equal to the sum of fuzzy dimensions
of fuzzy vector spaces.


[1] K. S. Abdulkhalikov, The dual of a fuzzy subspace, Fuzzy Sets and Systems, 82 (1996),
[2] A. K. Katsaras and D. B. Liu, Fuzzy vector spaces and fuzzy topological vector spaces, J.
Math. Anal. Appl., 58 (1977), 135-146.
[3] R. Kumar, On the dimension of a fuzzy subspace, Fuzzy Sets and Systems, 54 (1993), 229-
[4] R. Lowen, Convex fuzzy sets, Fuzzy Sets and Systems, 3 (1980), 291-310.
[5] G. Lubczonok and V. Murali, On
ags and fuzzy subspaces of vector spaces, Fuzzy Sets and
Systems, 125 (2002), 201-207.
[6] P. Lubczonok, Fuzzy vector spaces, Fuzzy Sets and Systems, 38 (1990), 329-343.
[7] F. G. Shi, A new approach to the fuzzi cation of matroids, Fuzzy Sets and Systems, 160
(2009), 696-705.
[8] F. G. Shi and C. E. Huang, Fuzzy bases and the fuzzy dimension of fuzzy vector spaces,
Mathematical Communications, 15(2) (2010), 303-310.
[9] L. Wang and F. G. Shi, Characterization of L-fuzzifying matroids by L-fuzzifying closure
operators, Iranian Journal of Fuzzy Systems, 7(1) (2010), 47-58.
[10] L. A. Zadeh, A computational approach to fuzzy quanti ers in natural languages, Comput.
Math. Appl., 9 (1983), 149-184.