IJPAM: Volume 112, No. 1 (2017)




R. Uzbashy$^1$, A.A. Hanano$^2$, E. Koudsi$^3$
$^{1,2,3}$Department of Mathematics
Faculty of Science
Damascus University
Damascus, SYRIA


This paper studies the complete reducibility of representations of $L$-fuzzy groups. We define the comparable family of $L$-fuzzy spaces, and prove the following theorem: Let $\widetilde T_{\mu}(\mathcal{V})$ be a completely reducible representation of an $L$-fuzzy group $\mu$ in a $\mu$-space $\mathcal{V}$, and $\{\mathcal{W}_i\}_{i\in I}$ be a family of $\mu$-subspaces of $\mathcal{V}$. If the intersection $\bigcap_{i\in{I}}\mathcal W_i$ is positive, then every $ {\mathcal{W}_i} $ is completely reducible.

In addition, we introduce a criterion to validate the complete reducibility of $\widetilde{T}_\mu(\mathcal{V})$ as follows: If the representation $T:Supp\,\mu\rightarrow GL(Supp\,\mathcal{V})$ is completely reducible, then $\widetilde T_{\mu}(\mathcal{V})$ is completely reducible too. However, we prove that the inverse is not necessarily true, but it is satisfied when the set $ \bigcup_{i\in{I}}\mathcal W_i $ is comparable.

Finally, as a result, we show that every representation of $L$-fuzzy group in any $\mu$-space $\mathcal{V}$ over the field of complex numbers is always completely reducible.


Received: November 29, 2016
Revised: January 12, 2017
Published: January 26, 2017

AMS Classification, Key Words

AMS Subject Classification: 20N25, 20F29
Key Words and Phrases: $L$-fuzzy group, $L$-fuzzy space, $\mu$-space, completely reducible representation

Download Section

Download paper from here.
You will need Adobe Acrobat reader. For more information and free download of the reader, see the Adobe Acrobat website.


J. A. Goguen, submitted by L. Zadeh, $L$-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 $L$-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 $L$-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.

How to Cite?

DOI: 10.12732/ijpam.v112i1.9 How to cite this paper?

International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2017
Volume: 112
Issue: 1
Pages: 115 - 124

$L$-FUZZY GROUPS%22&as_occt=any&as_epq=&as_oq=&as_eq=&as_publication=&as_ylo=&as_yhi=&as_sdtAAP=1&as_sdtp=1" title="Click to search Google Scholar for this entry" rel="nofollow">Google Scholar; DOI (International DOI Foundation); WorldCAT.

CC BY This work is licensed under the Creative Commons Attribution International License (CC BY).