Hesitant sets and hesitant relations

Firstly, this paper is intended to point out the lack of mathematical rigor in logic operation definition of hesitant fuzzy sets and to revise these operations. Then, new logical operations and partially ordered relations are introduced on a linearly ordered set, whose properties are studied accordingly. Finally, the concepts of hesitant set and hesitant relation are given together with their operations and corresponding properties. It can be found that some existing concepts are special examples of hesitant sets, such as shadowed sets, hesitant fuzzy sets and interval-valued hesitant fuzzy sets etc.