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Fuzzy hypervector spaces. (English) Zbl 1211.15002
Summary: The aim of this paper is the generalization of the notion of fuzzy vector spaces to fuzzy hypervector spaces. In this regard, by considering the notion of fuzzy hypervector spaces, we characterize a fuzzy hypervector space based on its level sub-hyperspaces. The algebraic nature of fuzzy hypervector spaces under transformations is studied. Certain conditions are obtained under which a given fuzzy hypervector space can or cannot be realized as a union of two fuzzy hypervector spaces such that none is contained in the other. The construction of a fuzzy hypervector space generated by a given fuzzy subset of a hypervector space is given. The set of all fuzzy cosets of a fuzzy hypervector space is shown to be a hypervector space. Finally, a fuzzy quotient hypervector space is defined and an analogue of a consequence of the “fundamental theorem of homomorphisms” is obtained.

##### MSC:
 15A03 Vector spaces, linear dependence, rank
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##### References:
 [1] F. Marty, “Sur une generalization de la notion de groupe,” in Proceedings of the 8th Congress des Mathematiciens Scandinaves, pp. 45-49, Stockholm, Sweden, 1934. · Zbl 61.1014.03 [2] P. Corsini, Prolegomena of Hypergroup Theory, Aviani Editor, Udine, Italy, 2nd edition, 1993. · Zbl 0785.20032 [3] P. Corsini and V. Leoreanu, Applications of Hyperstructure Theory, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003. · Zbl 1027.20051 [4] T. Vougiuklis, Hyperstructures and Their Representations, Hadronic Press, Palm Harbor, Fla, USA, 1994. [5] M. S. Tallini, “Hypervector spaces,” in Proceeding of the 4th International Congress in Algebraic Hyperstructures and Applications, pp. 167-174, Xanthi, Greece, 1990. · Zbl 0801.20057 [6] M. S. Tallini, “Weak hypervector spaces and norms in such spaces,” in Algebraic Hyperstructures and Applications, pp. 199-206, Hadronic Press, Palm Harbor, Fla, USA, 1994. · Zbl 0840.15001 [7] L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, no. 3, pp. 338-353, 1965. · Zbl 0139.24606 · doi:10.1016/S0019-9958(65)90241-X [8] A. Rosenfeld, “Fuzzy groups,” Journal of Mathematical Analysis and Applications, vol. 35, no. 3, pp. 512-517, 1971. · Zbl 0194.05501 · doi:10.1016/0022-247X(71)90199-5 [9] S. Nanda, “Fuzzy linear spaces over valued fields,” Fuzzy Sets and Systems, vol. 42, no. 3, pp. 351-354, 1991. · Zbl 0738.15003 · doi:10.1016/0165-0114(91)90113-5 [10] A. K. Katsaras and D. B. Liu, “Fuzzy vector spaces and fuzzy topological vector spaces,” Journal of Mathematical Analysis and Applications, vol. 58, no. 1, pp. 135-146, 1977. · Zbl 0358.46011 · doi:10.1016/0022-247X(77)90233-5 [11] R. Ameri, “Fuzzy hypervector spaces over valued fields,” Iranian Journal of Fuzzy Systems, vol. 2, no. 1, pp. 37-47, 2005. · Zbl 1112.15001 [12] R. Ameri, “Fuzzy (Co-)norm hypervector spaces,” in Proceeding of the 8th International Congress in Algebraic Hyperstructures and Applications, pp. 71-79, Samotraki, Greece, September 2002. · Zbl 1036.15001 [13] R. Ameri and M. M. Zahedi, “Hypergroup and join spaces induced by a fuzzy subset,” Pure Mathematics and Applications, vol. 8, pp. 155-168, 1997. · Zbl 0905.20050 [14] R. Ameri and M. M. Zahedi, “Fuzzy subhypermodules over fuzzy hyperrings,” in Proceedings of the 6th International Congress in Algebraic Hyperstructures and Applications, pp. 1-14, Democritus University, Prague, Czech Republic, September 1996. · Zbl 0883.16037 [15] P. Corsini and V. Leoreanu, “Fuzzy sets and join spaces associated with rough sets,” Rendiconti del Circolo Matematico di Palermo, vol. 51, no. 3, pp. 527-536, 2002. · Zbl 1176.03035 · doi:10.1007/BF02871859 [16] P. Corsini and I. Tofan, “On fuzzy hypergroups,” Pure Mathematics and Applications, vol. 8, no. 1, pp. 29-37, 1997. · Zbl 0906.20049 [17] B. Davvaz, “Fuzzy Hv-submodules,” Fuzzy Sets and Systems, vol. 117, no. 3, pp. 477-484, 2001. · Zbl 0974.16041 · doi:10.1016/S0165-0114(98)00366-2 [18] B. Davvaz, “Fuzzy Hv-groups,” Fuzzy Sets and Systems, vol. 101, no. 1, pp. 191-195, 1999. · Zbl 0935.20065 · doi:10.1016/S0165-0114(97)00071-7 [19] R. Ameri and O. R. Dehghan, “On dimension of hypervector spaces,” to appear in European Journal in Mathematics. · Zbl 1157.15033