Decomposition theorems of fuzzy soft sets with finite value spaces Feng Feng a,∗ , Hamido Fujita b , Young Bae Jun c , Madad Khan d a Department of Applied Mathematics, School of Science, Xi’an University of Posts and Telecommunications, Xi’an 710121, China b Faculty of Software and Information Science, Iwate Prefectural University, 020-0193 Iwate, Japan c Department of Mathematics Education (and RINS), Gyeongsang National University, Chinju 660-701, Republic of Korea d Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad 22060, Pakistan Abstract The notion of fuzzy soft sets is a hybrid soft computing model that integrates both gradualness and parametrization 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 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. Key words: Fuzzy soft sets; Level soft sets; Lattice; Value space; Decomposition theorems ∗ Corresponding author. Tel.: +86 29 88166086. Email addresses: fengnix@hotmail.com (Feng Feng), HFujita-799@acm.org (Hamido Fujita), skywine@gmail.com (Young Bae Jun), madadmath@yahoo.com (Madad Khan). Preprint submitted to SCIENTIFIC WORLD JOURNAL 5 December 2013