On Neutrosophic Triplet Groups: Basic Properties, NT-Subgroups, and Some Notes

Zhang, Xiao­hong; Hu, Qingqing; Smarandache, Florentín; An, Xiaogang

doi:10.3390/sym10070289

Cited by 25 publications

(40 citation statements)

References 17 publications

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“…Related theories of neutrosophic triplet, duplet, and duplet set were developed by Smarandache [18]. Neutrosophic duplets and triplets have fascinated several researchers who have developed concepts such as neutrosophic triplet normed space, fields, rings and their applications; triplets cosets; quotient groups and their application to mathematical modeling; triplet groups; singleton neutrosophic triplet group and generalization; and so on [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36]. Computational and combinatorial aspects of algebraic structures are analyzed in [37].…”

Section: Introductionmentioning

confidence: 99%

Neutrosophic Triplets in Neutrosophic Rings

2019

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The neutrosophic triplets in neutrosophic rings Q ∪ I and R ∪ I are investigated in this paper. However, non-trivial neutrosophic triplets are not found in Z ∪ I . In the neutrosophic ring of integers Z \ {0, 1}, no element has inverse in Z. It is proved that these rings can contain only three types of neutrosophic triplets, these collections are distinct, and these collections form a torsion free abelian group as triplets under component wise product. However, these collections are not even closed under component wise addition.

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Section: Introductionmentioning

confidence: 99%

Neutrosophic Triplets in Neutrosophic Rings

2019

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“…Single-valued neutrosophic sets were successfully applied to various decision making problems [3][4][5][6][7][8]. In addition, Zhang et al studied the neutrosophic logic algebras and discussed neutrosophic filters and neutrosophic triplet groups, which are the important foundation of the development of neutrosophic logic theory [9][10][11][12]. To facilitate research, "single-valued neutrosophic sets" are abbreviated as "neutrosophic sets" in this paper.…”

Section: Introductionmentioning

confidence: 99%

Neutrosophic Triangular Norms and Their Derived Residuated Lattices

Zhang

2019

Symmetry

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Neutrosophic triangular norms (t-norms) and their residuated lattices are not only the main research object of neutrosophic set theory, but also the core content of neutrosophic logic. Neutrosophic implications are important operators of neutrosophic logic. Neutrosophic residual implications based on neutrosophic t-norms can be applied to the fields of neutrosophic inference and neutrosophic control. In this paper, neutrosophic t-norms, neutrosophic residual implications, and the residuated lattices derived from neutrosophic t-norms are investigated deeply. First of all, the lattice and its corresponding system are proved to be a complete lattice and a De Morgan algebra, respectively. Second, the notions of neutrosophic t-norms are introduced on the complete lattice discussed earlier. The basic concepts and typical examples of representable and non-representable neutrosophic t-norms are obtained. Naturally, De Morgan neutrosophic triples are defined for the duality of neutrosophic t-norms and neutrosophic t-conorms with respect to neutrosophic negators. Third, neutrosophic residual implications generated from neutrosophic t-norms and their basic properties are investigated. Furthermore, residual neutrosophic t-norms are proved to be infinitely ∨-distributive, and then some important properties possessed by neutrosophic residual implications are given. 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. T-norms were originally defined by Menger [20], and then Schweizer and Sklar [21,22] redefined the t-norms which have been used to today. From the perspective of fuzzy logic, t-norms are the extension of intersection operation of fuzzy sets [23]. The algebraic properties of t-norms, for example, continuity, archimedean, strict, nilpotent and so on, are discussed in some papers [24][25][26][27][28]. Hu et al. studied t-norm extension operations [29]. Wang et al. discussed the lattice structure of algebra of fuzzy values [30]. The t-norms which satisfy the residual principle are an important class of t-norms, because they can produce fuzzy implications and constitute the residuated lattices [31][32][33][34]. Type-2 t-norms (t-conorms) and their residual operators on type-2 fuzzy sets were investigated by Li [25], which promote the development of fuzzy reference system. Intuitionistic fuzzy t-norms on intuitionistic fuzzy sets (L * -fuzzy sets) were proposed by Deschrij...

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“…For the nature and structure of NTG, recently, some new results have been published: cancellable NTGs are discussed in [17]; several homomorphism theorems of commutative NTGs are proved in [18]; some new properties of NTGs are obtained in [19]; the relationships between generalized NTGs and logical algebraic systems are investigated in [20]. In these papers, the name "neutrosophic triplet group" essentially refers to "neutrosophic extended triplet group", which is illustrated by the authors of [17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%

On Homomorphism Theorem for Perfect Neutrosophic Extended Triplet Groups

et al. 2018

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Some homomorphism theorems of neutrosophic extended triplet group (NETG) are proved in the paper [Fundamental homomorphism theorems for neutrosophic extended triplet groups, Symmetry 2018, 10(8), 321; doi:10.3390/sym10080321]. These results are revised in this paper. First, several counterexamples are given to show that some results in the above paper are not true. Second, two new notions of normal NT-subgroup and complete normal NT-subgroup in neutrosophic extended triplet groups are introduced, and their properties are investigated. Third, a new concept of perfect neutrosophic extended triplet group is proposed, and the basic homomorphism theorem of perfect neutrosophic extended triplet groups is established.

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On Neutrosophic Triplet Groups: Basic Properties, NT-Subgroups, and Some Notes

Cited by 25 publications

References 17 publications

Neutrosophic Triplets in Neutrosophic Rings

Neutrosophic Triplets in Neutrosophic Rings

Neutrosophic Triangular Norms and Their Derived Residuated Lattices

On Homomorphism Theorem for Perfect Neutrosophic Extended Triplet Groups

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