New Operations of Totally Dependent-Neutrosophic Sets and Totally Dependent-Neutrosophic Soft Sets

2019

Mathematics

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In this paper, some new properties of Abel Grassmann‘s Neutrosophic Extended Triplet Loop (AG-NET-Loop) were further studied. The following important results were proved: (1) an AG-NET-Loop is weakly commutative if, and only if, it is a commutative neutrosophic extended triplet (NETG); (2) every AG-NET-Loop is the disjoint union of its maximal subgroups. At the same time, the new notion of Abel Grassmann’s (l, l)-Loop (AG-(l, l)-Loop), which is the Abel-Grassmann’s groupoid with the local left identity and local left inverse, were introduced. The strong AG-(l, l)-Loops were systematically analyzed, and the following decomposition theorem was proved: every strong AG-(l, l)-Loop is the disjoint union of its maximal sub-AG-groups.

Section: Introductionmentioning

confidence: 99%

The Decomposition Theorems of AG-Neutrosophic Extended Triplet Loops and Strong AG-(l, l)-Loops

2019

Mathematics

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“…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.…”

Section: Introductionmentioning

confidence: 99%

Neutrosophic Triangular Norms and Their Derived Residuated Lattices

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...

“…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

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: