Study on the Algebraic Structure of Refined Neutrosophic Numbers

Abstract

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: