Three-way decision spaces based on partially ordered sets and three-way decisions based on hesitant fuzzy sets

Three-way decisions on three-way decision spaces are based on fuzzy lattices, i.e. complete distributive lattices with involutive negators. However, now some popular structures, such as hesitant fuzzy sets and type-2 fuzzy sets, do not constitute fuzzy lattices. It limits applications of the theory of three-way decision spaces. So this paper attempts to generalize measurement on decision conclusion in three-way decision spaces from fuzzy lattices to partially ordered sets. First three-way decision spaces and three-way decisions are discussed based on general partially ordered sets. Then this paper points out that the collection of non-empty subset of [0, 1] and the family of hesitant fuzzy sets are both partially ordered sets. Finally this paper systematically discusses three-way decision spaces and three-way decisions based on hesitant fuzzy sets and interval-valued hesitant fuzzy sets and obtains many useful decision evaluation functions.