IJPAM: Volume 112, No. 1 (2017)

Title

COMPLETE REDUCIBILITY OF
REPRESENTATIONS OF $L$-FUZZY GROUPS

Authors

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

Abstract

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.

History

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

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How to Cite?

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

Source:
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


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