Decomposition theorems and extension principles for hesitant fuzzy sets

Every typical HFS is represented by a well-structured family of FSs.We define uniformly typical HFS and characteristic of a HFS.We define new families of cuts for HFSs.We prove the first decomposition theorem of HFSs in the literature.Two extension principles extend crisp maps to maps between HFSs. We prove a decomposition theorem for hesitant fuzzy sets, which states that every typical hesitant fuzzy set on a set can be represented by a well-structured family of fuzzy sets on that set. This decomposition is expressed by the novel concept of hesitant fuzzy set associated with a family of hesitant fuzzy sets, in terms of newly defined families of their cuts. Our result supposes the first representation theorem of hesitant fuzzy sets in the literature. Other related representation results are proven. We also define two novel extension principles that extend crisp functions to functions that map hesitant fuzzy sets into hesitant fuzzy sets.