Abstract
Fuzzy soft sets realize a hybrid soft computing model in which the methods of gradualness and parametrization for dealing with uncertainty are combined effectively. Up to now, fuzzy soft sets have shown to be useful in various fields such as algebra, logic, data mining, supply chains risk management, forecasting, prediction and decision making under uncertainty. Also it is well-known that the level set approach and decomposition theorems play an important role in investigating fuzzy concepts or structures. Thus decomposition of fuzzy soft sets is a meaningful research topic from both theoretical and practical viewpoints.
Nevertheless, so far as we know very few works contribute to this important issue. The present study endeavors to fill this blank in the theory of fuzzy soft sets. It offers a systematic investigation of scalar products of fuzzy soft sets. Particularly it is shown that scalar product operations can be regarded as semimodule actions and algebraic structures like ordered idempotent semimodules of fuzzy soft sets over ordered semirings can be constructed. Also the collection of all t-level soft sets of a fuzzy soft set can form a distributive lattice under soft union and intersection operations. Finally, some decomposition theorems for fuzzy soft sets are established using scalar products and level soft sets with either constant thresholds or variable thresholds given by fuzzy sets.