Ameri, R.; Dehghan, O. Fuzzy hypervector spaces based on fuzzy singletons. (English) Zbl 1222.03057 Comput. Math. Appl. 61, No. 10, 2933-2943 (2011). Summary: We introduce and study fuzzy hypervector spaces based on fuzzy singletons. In this regard, by considering the notion of fuzzy singletons, we characterize a fuzzy hypervector space fuzzily spanned by a fuzzy subset. Then we use these results to introduce the concept of fuzzy freeness of a fuzzy subset \(\mu \) of a hypervector space \(V\) and finally we characterize it in terms of linear independence in the usual sense. Cited in 4 Documents MSC: 03E72 Theory of fuzzy sets, etc. 15A03 Vector spaces, linear dependence, rank, lineability Keywords:fuzzy hypervector space; fuzzy singleton; fuzzy spanning; fuzzy freeness PDFBibTeX XMLCite \textit{R. Ameri} and \textit{O. Dehghan}, Comput. Math. Appl. 61, No. 10, 2933--2943 (2011; Zbl 1222.03057) Full Text: DOI References: [1] F. Marty, Sur une generalization de la notion de groupe, in: 8th Congress des Mathematiciens Scandinaves, Stockholm, 1934, pp. 45-49.; F. Marty, Sur une generalization de la notion de groupe, in: 8th Congress des Mathematiciens Scandinaves, Stockholm, 1934, pp. 45-49. · JFM 61.1014.03 [2] Corsini, P., Prolegomena of Hypergroup Theory (1993), Aviani editor · Zbl 0785.20032 [3] Corsini, P.; Leoreanu, V., Applications of Hyperstructure Theory (2003), Kluwer Academic Publications · Zbl 1027.20051 [4] Vougiuklis, T., Hyperstructures and their Representations (1994), Hardonic Press [5] M.S. Tallini, Hypervector spaces, in: Fourth Int. Congress on AHA, 1990, pp. 167-174.; M.S. Tallini, Hypervector spaces, in: Fourth Int. Congress on AHA, 1990, pp. 167-174. · Zbl 0801.20057 [6] Tallini, M. S., Weak hypervector spaces and norms in such spaces, (Algebraic Hyperstructures and Applications (1994), Hardonic Press), 199-206 · Zbl 0840.15001 [7] Zadeh, L. A., Fuzzy sets, Information and Control, 8, 338-353 (1965), Generalized fuzzy hyperideals of hyperrings purchase the full-text article · Zbl 0139.24606 [8] Rosenfeld, A., Fuzzy groups, Journal of Mathematical Analysis and Applications, 35, 512-517 (1971) · Zbl 0194.05501 [9] Nanda, S., Fuzzy linear spaces over valued fields, Fuzzy Sets and Systems, 42, 351-354 (1991) · Zbl 0738.15003 [10] Katsaras, A. K.; Liu, D. B., Fuzzy vector spaces and fuzzy topological vector spaces, Journal of Mathematical Analysis and Applications, 58, 135-146 (1977) · Zbl 0358.46011 [11] Malik, D. S.; Mordeson, J. N., Fuzzy vector space, Information Science, 55, 271-281 (1991) · Zbl 0727.15001 [12] Mordeson, J. N., Generating properties of fuzzy algebraic structures, Fuzzy Sets and Systems, 55, 107-120 (1994) · Zbl 0784.12007 [13] Kumar, R., Fuzzy vector space and fuzzy cosets, Fuzzy Sets and Systems, 45, 109-116 (1992) · Zbl 0747.15001 [14] Ameri, R., Fuzzy hypervector spaces over valued fields, Iranian Journal of Fuzzy Systems, 2, 37-47 (2005) · Zbl 1112.15001 [15] R. Ameri, Fuzzy (Co-)norm hypervector spaces, in: Proceeding of the 8th International Congress in Algebraic Hyperstructures and Applications, Samotraki, Greece, September 1-9, 2002, pp. 71-79.; R. Ameri, Fuzzy (Co-)norm hypervector spaces, in: Proceeding of the 8th International Congress in Algebraic Hyperstructures and Applications, Samotraki, Greece, September 1-9, 2002, pp. 71-79. · Zbl 1036.15001 [16] R. Ameri, O.R. Dehghan, Fuzzy hypervector spaces, Advances in Fuzzy Systems, vol. 2008, Article ID 295649.; R. Ameri, O.R. Dehghan, Fuzzy hypervector spaces, Advances in Fuzzy Systems, vol. 2008, Article ID 295649. · Zbl 1211.15002 [17] Ameri, R.; Zahedi, M. M., Hypergroup and join spaces induced by a fuzzy subset, Pure Mathematics and Applications. PU.M.A., 8, 155-168 (1997) · Zbl 0905.20050 [18] Corsini, P.; Tofan, I., On fuzzy hypergroups, Pure Mathematics and Applications. PU.M.A., 8, 29-37 (1997) · Zbl 0906.20049 [19] Davvaz, B., Fuzzy \(H_V\)-submodules, Fuzzy Sets and Systems, 117, 477-484 (2001) · Zbl 0974.16041 [20] Davvaz, B., Fuzzy \(H_V\)-groups, Fuzzy Sets and Systems, 101, 191-195 (1999) · Zbl 0935.20065 [21] Zhana, J.; Davvaz, B.; Shum, K. P., Generalized fuzzy hyperideals of hyperrings, Computers and Mathematics with Applications, 56, 7, 1732-1740 (2008) · Zbl 1152.16040 [22] Ameri, R.; Dehghan, O. R., Fuzzy basis of fuzzy hypervector spaces, Iranian Journal of Fuzzy Systems, 7, 97-213 (2010) · Zbl 1251.15001 [23] Ameri, R.; Dehghan, O. R., On dimension of hypervector spaces, European Journal of Pure and Applied Mathematics, 1, 2, 32-50 (2008) · Zbl 1157.15033 [24] R. Ameri, O.R. Dehghan, Dimension of fuzzy hypervector spaces, Iranian Journal of Fuzzy Systems (in press).; R. Ameri, O.R. Dehghan, Dimension of fuzzy hypervector spaces, Iranian Journal of Fuzzy Systems (in press). · Zbl 1260.15045 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.