IJPAM: Volume 112, No. 1 (2017)
REPRESENTATIONS OF -FUZZY GROUPS , A.A. Hanano, E. Koudsi
Department of Mathematics
Faculty of Science
-fuzzy groups. We define the comparable family of -fuzzy spaces, and prove the following theorem: Let be a completely reducible representation of an -fuzzy group in a -space , and be a family of -subspaces of . If the intersection is positive, then every is completely reducible.
In addition, we introduce a criterion to validate the complete reducibility of as follows: If the representation is completely reducible, then is completely reducible too. However, we prove that the inverse is not necessarily true, but it is satisfied when the set is comparable.
Finally, as a result, we show that every representation of -fuzzy group in any -space over the field of complex numbers is always completely reducible.
Received: November 29, 2016
Revised: January 12, 2017
Published: January 26, 2017
AMS Subject Classification: 20N25, 20F29
Key Words and Phrases: -fuzzy group, -fuzzy space, -space, completely reducible representation
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- J. A. Goguen, submitted by L. Zadeh, -fuzzy Sets, Journal of Mathematical Analysis and Applications, 18, No. 1 (1967), 145-174, doi: https://doi.org/10.1016/0022-247X(67)90189-8.
- D. B. Liu, A. K. Katsaras, Fuzzy vector spaces and fuzzy topological vector spaces, Journal of Mathematics Analysis and Applications, 58, No. 1 (1977), 135-146, doi: https://doi.org/10.1016/0022-247X(77)90233-5.
- P. Lubczonok, Fuzzy vector spaces, Fuzzy Sets and System, 38, No. 3 (1990), 329-343, doi: https://doi.org/10.1016/0165-0114(90)90206-L.
- J.N. Mordeson, P.S. Nair, Fuzzy Mathematics: An Introduction for Engineers and Scientists, 20 of Janusz Kacprzyk, 20 of Studies in fuzziness and soft computing Springer-Verlag company (2001), 89-112, doi: https://doi.org/10.1007/978-3-7908-1808-6.
- J.N. Mordeson, K.R. Bhutani, A. Rosenfeld, Fuzzy Group Theory, 182 of Studies in Fuzziness and Soft Computing, Springer Berlin Heidelberg (2005), 344-350, doi: https://doi.org/10.1007/b12359.
- S. Ovchinnikov, On the Image of an -Fuzzy Group, Mathematics Department, San Francisco State University, 94, Issue 1 (1998), 129-131, doi: https://doi.org/10.1016/S0165-0114(96)00361-2.
- A. Rosenfeld, Fuzzy groups, Journal of Mathematics Analysis and Applications, 35, No. 3 (1971), 512-517, doi: https://doi.org/10.1016/0022-247X(71)90199-5.
- R. Uzbashy, A.A. Hanano, E. Koudsi, An Extension of Maschke's Theorem to the Representations of -Fuzzy Groups, Mathematics Department, Damascus University, Damascus, Syria (2015), In Press.
- L.A. Zadeh, Fuzzy sets, Information and Control, 8, No. 3 (1965), 338-353, doi: https://doi.org/10.1016/S0019-9958(65)90241-X.
Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Pages: 115 - 124
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