Abstract
In this paper, properties of operations and algebraic structures of hesitant fuzzy sets are investigated. Semilattices of hesitant fuzzy sets with union and intersection are discussed, respectively. By using ⊕ and ⊗ operators, the commutative monoid of hesitant fuzzy sets is provided, moreover, the lattice and distributive lattice of hesitant fuzzy sets are defined on the equivalence class of hesitant fuzzy sets. Based on the distributive lattice of hesitant fuzzy sets, the residuated lattices of hesitant fuzzy sets are constructed by residual implications, which are induced by intersection and ⊗, respectively. From the theoretical point of view, algebraic structures of hesitant fuzzy sets are useful for approximate reasoning and decision making to deal with hesitancy of information.
Highlights
Because of various types of uncertainties present in economics, engineering and decision making, theories of probability, fuzzy set [30] and rough set [13] as well-known and often useful mathematical tools have been proposed to describe and handle those uncertainties
Hesitant fuzzy sets (HFSs) [19, 20] and its applications are progressing rapidly [18], as a generalization of fuzzy sets, hesitant fuzzy sets (HFSs) are more suitable for dealing with the situations where decision makers have hesitancy in providing their preferences over objects, rather than a margin of error considered in intuitionistic fuzzy sets (IFSs) [1] or some possibility distribution on the possible values considered in type-2 fuzzy sets (T2FS) and type-n fuzzy sets (T-nFSs) [5, 12]
There are some conclusions of HFSs show that they are different to IFSs because the envelope of HFSs can be considered as an IFS characterized by a membership degree and a non-membership degree, different to T-2FSs [11,27] because all HFSs are T-2FSs in which the membership degree of a given element is defined as a fuzzy set, different to fuzzy multisets (FMs) because HFSs and FMs are of the same form but have different operations [19, 20]
Summary
Because of various types of uncertainties present in economics, engineering and decision making, theories of probability, fuzzy set [30] and rough set [13] as well-known and often useful mathematical tools have been proposed to describe and handle those uncertainties. Confidence levels (or degrees) are used in all extension of fuzzy sets, such as, in [29], many intuitionistic fuzzy aggregation operators with confidences levels of aggregated arguments are proposed and utilized in multiple attribute group decision making problems with intuitionistic fuzzy information. Inspired by existed interesting conclusions of fuzzy sets and fuzzy decision making with confidence levels, we investigate properties of operations on HFSs with α−confidence level and algebraic structures of hesitant fuzzy sets in this paper. Based on distributive lattices of hesitant fuzzy sets, residuated lattices of hesitant fuzzy sets are constructed by residual implications, which are induced by intersection or ⊗ with α−confidence level, respectively.
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