Neutrosophic Duplet Semi-Group and Cancellable Neutrosophic Triplet Groups

Zhang, Xiaohong; Smarandache, Florentín; Liang, Xingliang

doi:10.3390/sym9110275

Cited by 47 publications

(52 citation statements)

References 14 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%

“…Computational and combinatorial aspects of algebraic structures are analyzed in [37]. Neutrosophic duplet semigroup [23], classical group of neutrosophic triplet groups [27], the neutrosophic triplet group [12], and neutrosophic duplets of {Z pn , ×} and {Z pq , ×} have been analyzed [28]. Thus, Neutrosophic triplets in case of the modulo integers Z n (2 < n < ∞) have been extensively researched [27].…”

Section: Introductionmentioning

confidence: 99%

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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%

Section: Introductionmentioning

confidence: 99%

Neutrosophic Triplets in Neutrosophic Rings

2019

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“…Moreover, some research papers have carried out in-depth research based on NTG (NETG). For example, the inclusion relations of neutrosophic sets [5], neutrosophic triplet coset [6], neutrosophic duplet semi-groups [7], generalized neutrosophic extended triplet group [8], AG-neutrosophic extended triplet loops [9,10], the neutrosophic set theory to pseudo-BCI algebras [11], neutrosophic triplet ring and a neutrosophic triplet field [12,13], neutrosophic triplet normed space [14], neutrosophic soft sets [15], neutrosophic vector spaces [16] and so on have been studied.…”

Section: Introductionmentioning

confidence: 99%

Study on the Algebraic Structure of Refined Neutrosophic Numbers

Zhang

et al. 2019

Symmetry

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This paper aims to explore the algebra structure of refined neutrosophic numbers. Firstly, the algebra structure of neutrosophic quadruple numbers on a general field is studied. Secondly, The addition operator ⊕ and multiplication operator ⊗ on refined neutrosophic numbers are proposed and the algebra structure is discussed. We reveal that the set of neutrosophic refined numbers with an additive operation is an abelian group and the set of neutrosophic refined numbers with a multiplication operation is a neutrosophic extended triplet group. Moreover, algorithms for solving the neutral element and opposite elements of each refined neutrosophic number are given.Symmetry 2019, 11, 954 2 of 13 neutrosophic quadruple number, which has form: NQ = a + bT + cI + dF where a, b, c, d are real (or complex) numbers; and T is the truth/membership/probability; I is the indeterminacy; and F is the false/membership/improbability are called Neutrosophic Quadruple (Real, respectively, Complex) Numbers. "a" is called the known part of NQ, while bT + cI + dF is called the unknown part of NQ. Similar to refined neutrosophic numbers, if T can be split into many types of truths, T 1 , T 2 , · · · , T p , I into many types of indeterminacies, I 1 , I 2 , · · · , I r , and F into many types of falsities, F 1 , F 2 , · · · , F r , we can get the refined neutrosophic quadruple numbers. We know that the set of neutrosophic quadruple numbers with a multiplication operation is a NETG. In this paper, we explore the algebra structure of refined neutrosophic numbers (refined neutrosophic quadruple numbers) and give new examples of NETG. In fact, the solving method of the neutral element and opposite elements for each refined neutrosophic number is different from the solving method for each neutrosophic quadruple number.The paper is organized as follows. Section 2 gives the basic concepts. In Section 3, we show that the set of neutrosophic quadruple numbers on the general field with a multiplication operation also consists of a NETG. In Section 4, the algebra structure of refined neutrosophic numbers and refined neutrosophic quadruple numbers are studied. Finally, the summary and future work is presented in Section 5. Basic ConceptsIn this section, we provide the related basic definitions and properties of NETG, neutrosophic quadruple numbers, and refined neutrosophic numbers (for details, see [3,4,[18][19][20]).Definition 1 ([3,4]). Let N be a non-empty set together with a binary operation * . Then, N is called a neutrosophic extended triplet set if, for any a ∈ N, there exists a neutral of "a" (denote by neut(a)), and an opposite of "a"(denote by anti(a)), such that neut(a) ∈ N, anti(a) ∈ N and: a * neut(a) = neut(a) * a = a, a * anti(a) = anti(a) * a = neut(a).The triplet (a, neut(a), anti(a)) is called a neutrosophic extended triplet.Definition 2 ([3,4]). Let (N, * ) be a neutrosophic extended triplet set. Then, N is called a neutrosophic extended triplet group (NETG), if the following conditions are satisfied:

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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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Neutrosophic Duplet Semi-Group and Cancellable Neutrosophic Triplet Groups

Cited by 47 publications

References 14 publications

Neutrosophic Triplets in Neutrosophic Rings

Neutrosophic Triplets in Neutrosophic Rings

Study on the Algebraic Structure of Refined Neutrosophic Numbers

On Homomorphism Theorem for Perfect Neutrosophic Extended Triplet Groups

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