A vague soft set is a combination of a vague set and a soft set. In this paper, four kinds of cut sets on vague soft sets are introduced, which are generations of cut sets on fuzzy sets and have the same properties as that of fuzzy sets. The relations among these kinds of cut sets are also discussed. Then, the definitions of order nested sets and inverse order nested sets are given. Furthermore, based on them, the decomposition theorems and representation theorems of vague soft sets are established.
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