Fuzzy Inner Product and Fuzzy Norm \\of Hyperspaces

Document Type : Research Paper


School of Mathematics, Statistics and Computer Science, College of Sciences, University of Tehran, P.O. Box 14155-6455, Teheran, Iran


We introduce and  study  fuzzy (co-)inner product and fuzzy
(co-)norm of hyperspaces. In this regard by considering  the notion
of hyperspaces, as a generalization of vector spaces, first we will
introduce the notion of fuzzy (co-)inner product in
 hyperspaces and will apply it to formulate the notions of
fuzzy (co-)norm and fuzzy (co-)orthogonality  in hyperspaces. In
particular, we will prove that to every fuzzy hyperspace  there is an associated
unique fuzzy inner product  in  a natural way.


\bibitem{A1} R. Ameri, {\it Fuzzy hypervector spaces over valued
fields,} Iranian Journal of Fuzzy Systems, \textbf{2} (2005), 37--47.

\bibitem{A2} R. Ameri and O. R. Dehghan,{\it On dimension of hypervector
spaces,} European Journal of Pure and Applied Mathematics,
\textbf{1}{\bf(2)} (2008), 32--50.

\bibitem{A3} R. Ameri and O. R. Dehghan,{\it fuzzy hypervector
spaces,} Advances in Fuzzy Systems, doi:10.1155/2008/295649
, Article ID 295649, 2008.

\bibitem{A5} R. Ameri and O. R. Dehghan,{\it fuzzy basis of fuzzy
hypervector spaces,}  Iranian Journal of Fuzzy Systems, \textbf{7}
(2010), 97--213.

\bibitem{A6}  R. Ameri and O. R. Dehghan, {\it fuzzy hypervector spaces based on fuzzy
singletons,} Computers and Mathematics with Applications, {\bf 61(10)}
  (2011),  2933--2943.

\bibitem{A7} R. Ameri and O. R. Dehghan, {\it dimension of fuzzy
hypervector spaces,} Iranian Journal of Fuzzy Systems, {\bf 8(5)}
(2011), 149--166.

\bibitem{A8} R. Ameri and M. M. Zahedi, {\it  Fuzzy Subhypermodules over fuzzy hyperrings},
Sixth International Congress on AHA, Democritus Univ., (1996),

\bibitem{A9} R. Ameri and M. M. Zahedi, {\it Hyperalgebraic system},
Italian Journal of Pure and Applied Mathematics,  \textbf{6} (1999),

\bibitem{A9} R. Ameri, \textit{Fuzzy (co-)norm hypervector spaces},
Proceeding of the 8th International Congress in Algebraic
Hyperstructures and Applications, Samotraki, Greece, September 1-9
(2002), 71-79.

\bibitem{C84}  D. S. Comer, {\it Polygroups derived from cogroups}, Journal
of Algebra,  \textbf{89} (1984), 397--405.

\bibitem{C79}  P. Corsini, {\it Prolegomena of hypergroup theory}, Aviani Editore,

\bibitem{CL03} P. Corsini and V. Leoreanu, {\it Applications of
hyperstructure theory}, Kluwer Academic Publications, 2003.

\bibitem{CL02}  P. Corsini and V. Leoreanu, {\it Fuzzy sets and Join Spaces
Associated with rough sets}, Rend. Circ. Mat., Palermo,  \textbf{51}
(2002), 527--536.

\bibitem{CT97} P. Corsini, I. Tofan, {\it On fuzzy hypergroups}, PU.M.A.,
 \textbf{8} (1997), 29--37.

\bibitem{D81}  P. S. Das, {\it Fuzzy groups and level subgroups}, J.
Math. Anal. Appl.,  \textbf{84} (1981), 264-269.

\bibitem{D81}  P. Das, {\it Fuzzy vector spaces under triangular norms}, Fuzzy Sets and Systems,
 \textbf{3} (1988), 73--85.

\bibitem{D99}  B. Davvaz {\it Fuzzy $H_v$-groups}, Fuzzy Sets and Systems,
 \textbf{101} (1999), 191-195.

\bibitem{D01}  B. Davvaz,{\it Fuzzy $H_v$- submodules}, Fuzzy Sets and Systems,
 \textbf{117} (2001) 477-484.

\bibitem{M34} D. S. Malik and J. N. Mordeson, {\it Fuzzy vector
spaces}, Information Sciences,  \textbf{55} (1991), 271--281.

\bibitem{M34}  F. Marty, {\it Sur une generalization de la notion de
groupe}, $8^{iem}$ congres des Mathematiciens Scandinaves,
Stockholm, (1934), 45--49.

\bibitem{N91} S. Nanda, {\it Fuzzy linear spaces over valued fields},
FSS,  \textbf{42} (1991), 351--354.

 \bibitem{N86} S. Nanda, {\it Fuzzy fields and fuzzy linear spaces}, Fuzzy Sets and Systems,  \textbf{19} (1986), 89--94.

\bibitem{R71}  A. Rosenfeld, {\it Fuzzy groups}, J. Math. Anal. Appl.,  \textbf{35} (1971), 512--517.

\bibitem{T90} M. S. Tallini, {\it Hypervector spaces}, Proceedings of
Fourth Int. Congress  in Algebraic  Hyperstructures and Applications, Xanthi, Greece, world scientific, (1990), 167--174.

\bibitem{T94} M. S. Tallini, {\it Hypervector spaces and norm in
such spaces}, Algebraic  Hyperstructures and Applications, Hardonic
Press, (1994), 199--206.

\bibitem{T97} G. Tallini, {\it Dimensions in multivalued algebraic
structures}, Italian Journal of Pure and Applied Mathematics,  \textbf{1}
(1997), 51--64.

\bibitem{T84} G. Tallini, {\it Geometric hypergroups and line spaces}, Acta Univ. Caroliae Math. Phys.,  \textbf{25} (1984), 51--64.

\bibitem{TX94} L. Tu and W. X. Gu, {\it  Fuzzy algebraic system (I):
Direct products}, Fuzzy Sets and Systems,  \textbf{61} (1994), 313--327.

\bibitem{V94} T. Vougiuklis, {\it Hyperstructures and their
representations}, Hardonic Press, Inc., 1994.

\bibitem{W37} H. S. Wall, {\it Hypergroups}, Amer. J. Math., (1937), 77--98.

\bibitem{WT92} G. Wenxiang and L. Tu, {\it Fuzzy linear spaces}, Fuzzy Sets and Systems,  \textbf{94} (1992), 377--380.

\bibitem{Z65} L. A. Zadeh, {\it Fuzzy sets,} Information and Control,  \textbf{8} (1965), 338--353.