New inclusion relation of neutrosophic sets with applications and related lattice structure

Abstract

The main purpose of this paper is to study the inclusion relations of neutrosophic sets and some applications in multiple attribute decision making. At first, the existing two definitions of inclusion relation (called type-1 and type-2 inclusion relation, respectively) of neutrosophic sets are analyzed, and the deficiencies of type-1 and type-2 inclusion relations are illustrated by examples (in fact, they are actually two extreme cases). Second, a new definition of inclusion relation of neutrosophic sets (call it type-3 inclusion relation) is introduced, and a new method of ranking of neutrosophic sets is given. The effectiveness of the ranking method is presented by some application examples in multiple attribute decision making. Finally, type-3 inclusion relation of neutrosophic sets and related lattice structure are investigated in a systematic way, the definitions of type-3 union and type-3 intersection operations are proposed, and the following important result is proved: all of neutrosophic sets based on a certainty domain constitute a generalized De Morgan algebra (non-distributive lattice) with respect to type-3 union, type-3 intersection and complement operations. From this, the essential difference between neutrosophic set and fuzzy set (and intuitionistic fuzzy set) is clarified theoretically.

Neutrosophic Triangular Norms and Their Derived Residuated Lattices

“…Proof. By Propositions 1-5 and the definition of the generalized De Morgan algebra [14,55], we can get that (D * ; ∧ 1 , ∨ 1 , c , 0 D * , 1 D * ) is a generalized De Morgan algebra. Furthermore, we can prove that (D * ;…”

“…Picture fuzzy sets proposed by Cuong [13] in 2013 is a direct So far, scholars have described the inclusion relation of neutrosophic sets from three different angles. The first definition is proposed by Smarandache [1,18,53] and denoted as ⊆ 1 ; the second one is mentioned in [2,14,54] and denoted by ⊆ 2 ; the third one is presented in [14,38,39,54] and denoted by ⊆ 3 . Furthermore, based on the correlation between union, intersection operations and inclusion relation, we can obtain three different types of union, intersection operations and their properties.…”

“…Finally, a method for producing neutrosophic t-norms from neutrosophic implications is presented, and the residuated lattices are constructed on the basis of neutrosophic t-norms and neutrosophic residual implications.Symmetry 2019, 11, 817 2 of 22 generalization of intuitionistic fuzzy sets, because their positive membership, neutral membership and negative membership are not independent completely. It is worth noting that picture fuzzy sets can be regarded as special neutrosophic sets [14,15], and also can be called standard neutrosophic sets [1,2,[16][17][18][19].Fuzzy logic plays a vital role in fuzzy set theory. T-norms, t-conorms, negators and implications are very important fuzzy logic operators.…”

“…Symmetry 2019, 11, 817 2 of 22 generalization of intuitionistic fuzzy sets, because their positive membership, neutral membership and negative membership are not independent completely. It is worth noting that picture fuzzy sets can be regarded as special neutrosophic sets [14,15], and also can be called standard neutrosophic sets [1,2,[16][17][18][19].…”

Single-Valued Neutrosophic Hesitant Fuzzy Choquet Aggregation Operators for Multi-Attribute Decision Making

“…However, without a specific description, it is difficult to apply the NS in real scientific and other areas. Therefore, some researchers proposed the interval neutrosophic set (INS), single-valued neutrosophic set (SVNS), multi-valued neutrosophic set (MVNS), and rough neutrosophic set (RNS) in [10][11][12][13][14], and studied their related properties in detail, among them, the membership functions of SVNSs are standard subsets of [0,1]. The study on aggregation operators is a practical subject, which has been received growing attention.…”