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The Scientific World Journal
Volume 2014, Article ID 902687, 10 pages
http://dx.doi.org/10.1155/2014/902687
Research Article

Decomposition of Fuzzy Soft Sets with Finite Value Spaces

1Department of Applied Mathematics, School of Science, Xi'an University of Posts and Telecommunications, Xi'an 710121, China
2Faculty of Software and Information Science, Iwate Prefectural University, Iwate 020-0193, Japan
3Department of Mathematics Education (and RINS), Gyeongsang National University, Jinju 660-701, Republic of Korea
4Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad 22060, Pakistan

Received 30 October 2013; Accepted 3 December 2013; Published 12 January 2014

Academic Editors: M. Akram, A. I. Ban, Y. Cao, I. Cristea, A. Croitoru, M. Finger, J. Mycka, and X.-p. Wang

Copyright © 2014 Feng Feng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The notion of fuzzy soft sets is a hybrid soft computing model that integrates both gradualness and parameterization methods in harmony to deal with uncertainty. The decomposition of fuzzy soft sets is of great importance in both theory and practical applications with regard to decision making under uncertainty. This study aims to explore decomposition of fuzzy soft sets with finite value spaces. Scalar uni-product and int-product operations of fuzzy soft sets are introduced and some related properties are investigated. Using t-level soft sets, we define level equivalent relations and show that the quotient structure of the unit interval induced by level equivalent relations is isomorphic to the lattice consisting of all t-level soft sets of a given fuzzy soft set. We also introduce the concepts of crucial threshold values and complete threshold sets. Finally, some decomposition theorems for fuzzy soft sets with finite value spaces are established, illustrated by an example concerning the classification and rating of multimedia cell phones. The obtained results extend some classical decomposition theorems of fuzzy sets, since every fuzzy set can be viewed as a fuzzy soft set with a single parameter.