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.

Highlights

  • The notion of neutrosophic set was proposed by F

  • Smarandache and Ali [2] proposed the algebraic system neutrosophic triplet group (NTG), which is an extension of the classical group but the neutral element is different from the classical algebraic unit element

  • We explore the algebra structure of refined neutrosophic numbers and give new examples of neutrosophic extended triplet group (NETG)

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Summary

Introduction

The notion of neutrosophic set was proposed by F. Smarandache [1], which is an extension of fuzzy set and in order to solve real-world problems. The idea of neutrosophic set has been applicable in related algebraic structures. Among these algebraic structures, Smarandache and Ali [2] proposed the algebraic system neutrosophic triplet group (NTG), which is an extension of the classical group but the neutral element is different from the classical algebraic unit element. To regard the unit element as a special neutral element, the neutrosophic extended triplet group (NETG) has been proposed [3,4] and the classical group is regarded as a special case of a NETG. 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

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