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Journal articles on the topic 'Neutrosophic quadruple number'

Author: Grafiati

Published: 3 June 2025

Last updated: 31 July 2025

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1

Alifviansyah, Kevin, and Agung Lukito. "BCK-Aljabar Kuadrupel Neutrosofik." MATHunesa: Jurnal Ilmiah Matematika 10, no. 1 (2022): 226–38. http://dx.doi.org/10.26740/mathunesa.v10n1.p226-238.

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Artikel ini memperkenalkan bilangan kuadrupel neutrosofik yang berbasis pada suatu himpunan untuk mengonstruksi BCK-aljabar kuadrupel neutrosofik. Kemudian diselidiki beberapa sifatnya dan akan dipelajari konsep ideal, ideal implikatif positif, ideal tertutup pada BCK-aljabar kuadrupel neutrosofik. Untuk subhimpunan dan dari BCK/BCI-aljabar kuadrupel neutrosofik, akan dikaji himpunan yang terdiri atas bilangan kuadrupel neutrosofik dengan syarat tertentu. Akan disajikan beberapa syarat agar himpunan menjadi ideal implikatif positif pada BCK-Aljabar kuadrupel neutrosofik dan himpunan menjadi id

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2

Jun, Young Bae, Seok-Zun Song, and Seon Jeong Kim. "Neutrosophic Quadruple BCI-Positive Implicative Ideals." Mathematics 7, no. 5 (2019): 385. http://dx.doi.org/10.3390/math7050385.

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By considering an entry (i.e., a number, an idea, an object, etc.) which is represented by a known part ( a ) and an unknown part ( b T , c I , d F ) where T , I , F have their usual neutrosophic logic meanings and a , b , c , d are real or complex numbers, Smarandache introduced the concept of neutrosophic quadruple numbers. Using the concept of neutrosophic quadruple numbers based on a set, Jun et al. constructed neutrosophic quadruple BCK/BCI-algebras and implicative neutrosophic quadruple BCK-algebras. The notion of a neutrosophic quadruple BCI-positive implicative ideal is introduced, and

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3

Li, Qiaoyan, Yingcang Ma, Xiaohong Zhang, and Juanjuan Zhang. "Neutrosophic Extended Triplet Group Based on Neutrosophic Quadruple Numbers." Symmetry 11, no. 5 (2019): 696. http://dx.doi.org/10.3390/sym11050696.

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In this paper, we explore the algebra structure based on neutrosophic quadruple numbers. Moreover, two kinds of degradation algebra systems of neutrosophic quadruple numbers are introduced. In particular, the following results are strictly proved: (1) the set of neutrosophic quadruple numbers with a multiplication operation is a neutrosophic extended triplet group; (2) the neutral element of each neutrosophic quadruple number is unique and there are only sixteen different neutral elements in all of neutrosophic quadruple numbers; (3) the set which has same neutral element is closed with respec

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4

Şahin, Memet, and Arif Sarıoğlan. "Neutrosophic Quadruple Metric Spaces." Symmetry 17, no. 7 (2025): 1096. https://doi.org/10.3390/sym17071096.

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Instead of measuring the distance between two points with a positive real number, determining the degree to which the distance between these two points is close, not close, or uncertain allows for more detailed measurement. Recently, researchers have overcome this grading problem by using probability distribution functions, along with fuzzy, intuitionistic fuzzy, and neutrosophic sets. This study pioneers neutrosophic quadruple metric spaces as a powerful new tool for quantifying distances under complex, multi-dimensional uncertainty. It provides a comprehensive mathematical structure, includi

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5

Li, Qiaoyan, Yingcang Ma, Xiaohong Zhang, and Juanjuan Zhang. "Study on the Algebraic Structure of Refined Neutrosophic Numbers." Symmetry 11, no. 8 (2019): 954. http://dx.doi.org/10.3390/sym11080954.

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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

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6

Iampan, Aiyared, Metawee Songsaeng, and Ghulam Muhiuddin. "Fuzzy Duplex UP-Algebras." European Journal of Pure and Applied Mathematics 13, no. 3 (2020): 459–71. http://dx.doi.org/10.29020/nybg.ejpam.v13i3.3752.

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Using the concept of a neutrosophic quadruple number to a fuzzy duplex number, we introduce the notion of a fuzzy duplex UP-algebra, and investigate some related properties. Also, we find the necessary condition for a fuzzy duplex UP-set to be a fuzzy duplex UP-algebra. Furthermore, we study the relationship between special subsets of a UP-algebra and special subsets of a fuzzy duplex UP-set.

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7

Muhiuddin, G., Ahmad Al-Kenani, Eun Roh, and Young Jun. "Implicative Neutrosophic Quadruple BCK-Algebras and Ideals." Symmetry 11, no. 2 (2019): 277. http://dx.doi.org/10.3390/sym11020277.

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A neutrosophic set is initiated by Smarandache, and it is a novel tool to deal with vagueness considering the truth, indeterminacy and falsity memberships satisfying the condition that their sum is less than 3. The concept of neutrosophic quadruple numbers was introduced by Florentin Smarandache. Using this idea, Jun et al. introduced the notion of neutrosophic quadruple B C K / B C I -numbers, and studied neutrosophic quadruple B C K / B C I -algebras. As a continuation of Jun et al.’s paper, the notion of implicative neutrosophic quadruple B C K -algebras is introduced, and several propertie

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8

W.B., Vasantha Kandasamy, Ilanthenral Kandasamy, and Florentin Smarandache. "Neutrosophic Quadruple Vector Spaces and Their Properties." Mathematics 7, no. 8 (2019): 758. http://dx.doi.org/10.3390/math7080758.

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In this paper authors for the first time introduce the concept of Neutrosophic Quadruple (NQ) vector spaces and Neutrosophic Quadruple linear algebras and study their properties. Most of the properties of vector spaces are true in case of Neutrosophic Quadruple vector spaces. Two vital observations are, all quadruple vector spaces are of dimension four, be it defined over the field of reals R or the field of complex numbers C or the finite field of characteristic p, Z p ; p a prime. Secondly all of them are distinct and none of them satisfy the classical property of finite dimensional vector s

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9

Merkepci, Hamiyet, and Ahmed Hatip. "Algorithms for Computing Pythagoras Triples and 4-Tiples in Some Neutrosophic Commutative Rings." International Journal of Neutrosophic Science 20, no. 3 (2023): 107–14. http://dx.doi.org/10.54216/ijns.200310.

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This paper is dedicated to study the number theoretical Pythagoras triples\4-tiples problem in several kinds of neutrosophic algebraic systems, where it finds an algorithm to find Pythagoras triples\4-tiples in commutative neutrosophic rings and refined neutrosophic rings too. Besides, the necessary and sufficient condition for a triple\4-tiple to be Pythagoras triple\4-tiple (quadruples) is obtained and proven in term of theorems. In addition, many numerical examples will be illustrated.

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10

V., Christianto1* F. Smarandache2 M. Aslam3. "How we can extend the standard deviation notion with neutrosophic interval and quadruple neutrosophic numbers." International Journal of Neutrosophic Science (IJNS) 2, no. 2 (2020): 72–76. https://doi.org/10.5281/zenodo.3728218.

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11

Memet, Şahin, and Kargın Abdullah. "Neutrosophic Triplet Group Based on Set Valued Neutrosophic Quadruple Numbers." December 10, 2019. https://doi.org/10.5281/zenodo.3569671.

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Smarandache introduced neutrosophic quadruple sets and neutrosophic quadruple numbers [45] in 2015. These sets and numbers are real or complex number valued. In this study, we firstly introduce set valued neutrosophic quadruple sets and numbers. We give some known and special operations for set valued neutrosophic quadruple numbers. Furthermore, Smarandache and Ali obtained neutrosophic triplet groups [30] in 2016. In this study, we firstly give neutrosophic triplet groups based on set valued neutrosophic quadruple number thanks to operations for set valued neutrosophic quadruple numbers. In t

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12

A., A. A. Agboola, Davvaz B., and Smarandache F. "Neutrosophic quadruple algebraic hyperstructures." July 12, 2017. https://doi.org/10.5281/zenodo.888880.

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The objective of this paper is to develop neutrosophic quadruple algebraic hyperstructures. Specically, we develop neutrosophic quadruple semihypergroups, neutrosophic quadruple canonical hypergroups and neutrosophic quadruple hyperrings and we present elementary properties which characterize them.

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13

Young, Bae Jun, Song Seok-Zun, Smarandache Florentin, and Bordbar Hashem. "Neutrosophic Quadruple BCK/BCI-Algebras." September 7, 2018. https://doi.org/10.5281/zenodo.1410851.

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The notion of a neutrosophic quadruple BCK/BCI-number is considered, and a neutrosophic quadruple BCK/BCI-algebra, which consists of neutrosophic quadruple BCK/BCI-numbers, is constructed. Several properties are investigated, and a (positive implicative) ideal in a neutrosophic quadruple BCK-algebra and a closed ideal in a neutrosophic quadruple BCI-algebra are studied.

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14

Young, Bae Jun, Song Seok-Zun, Smarandache Florentin, and Bordbar Hashem. "Neutrosophic Quadruple BCK/BCI-Algebras." May 16, 2019. https://doi.org/10.5281/zenodo.2862227.

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The notion of a neutrosophic quadruple BCK/BCI-number is considered, and a neutrosophic quadruple BCK/BCI-algebra, which consists of neutrosophic quadruple BCK/BCI-numbers, is constructed. Several properties are investigated, and a (positive implicative) ideal in a neutrosophic quadruple BCK-algebra and a closed ideal in a neutrosophic quadruple BCI-algebra are studied.

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15

M., Al-Tahan, and Davvaz B. "Refined neutrosophic quadruple (po-)hypergroups and their fundamental group." July 9, 2019. https://doi.org/10.5281/zenodo.3275550.

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After introducing the notion of hyperstructures about 80 years ago by F. Marty, a number of researches on its theory, generalization, and it’s applications have been done. On the other hand, the theory of Neutrosophy, the study of neutralities, was developed in 1995 by F. Smarandache as an extension of dialectics. This paper aims at finding a connection between refined neutrosophy of sets and hypergroups. In this regard, we define refined neutrosophic quadruple hypergroups, study their properties, and find their fundamental refined neutrosophic quadruple groups. Moreover, some results re

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16

G., Muhiuddin, Smarandache Florentin, and Bae Jun Young. "Neutrosophic quadruple ideals in neutrosophic quadruple BCI-algebras." May 16, 2019. https://doi.org/10.5281/zenodo.2862271.

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17

Madeleine, Al-Tahan, and Davvaz Bijan. "On Some Prop erties of Neutrosophic Q uadruple Hv -rings." October 4, 2020. https://doi.org/10.5281/zenodo.4065443.

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  Hyperstructure theory, an 86 years old theory, has been of great interest for many algebraists where their researches were divided in to two categories: theory and applications. On the other hand, neutrosophic theory which is the study of neutralities, was introduced and developed by F. Smarandache in 1995 as an extension of dialectics. The purpose of this paper is to study some connections between the two theories: Neutrosophy and hyperstructures. In this regard, we define neutrosophic quadruple <em>H</em><em>v</em>-rings, neutrosophic quadruple <em>H</em><em>v</em>-subrings, and neutr

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18

G., Muhiuddin, Smarandache Florentin, and Bae Jun Young. "Neutrosophic quadruple ideals in neutrosophic quadruple BCI-algebras." April 6, 2019. https://doi.org/10.5281/zenodo.2631518.

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In the present paper, we discuss the Neutrosophic quadruple q-ideals and (regular) neutrosophic quadruple ideals and investigate their related properties. Also, for any two nonempty subsets U and V of a BCI-algebra S, conditions for the set NQ(U; V ) to be a (regular) neutrosophic quadruple ideal and a neutrosophic quadruple q-ideal of a neutrosophic quadruple BCI-algebra NQ(S) are discussed. Furthermore, we prove that let U; V; I and J be ideals of a BCI-algebra S such that I U and J V . If I and J are q-ideals of S, then the neutrosophic quadruple (U; V )-set NQ(U; V ) is a neutrosophic quad

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19

Vasantha, Kandasamy, Kandasamy Ilanthenral, and Smarandache Florentin. "Neutrosophic Quadruple Algebraic Codes over Z2 and their properties." May 3, 2020. https://doi.org/10.5281/zenodo.3782891.

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In this paper we for the rst time develop, dene and describe a new class of algebraic codes using Neutrosophic Quadruples which uses the notion of known value, and three unknown triplets (T; I; F) where T is the truth value, I is the indeterminate and F is the false value. Using this Neutrosophic Quadruples several researchers have built groups, NQ-semigroups, NQ-vector spaces and NQ linear algebras. However, so far NQ algebraic codes have not been developed or dened. These NQ-codes have some peculiar properties like the number of message symbols are always xed as 4-tuples, that is why we call

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20

Madeleine, Al Tahan, Davvaz Bijan, Smarandache Florentin, and Al-Kaseasbeh Saba. "Fundamental group and complete parts of Neutrosophic Quadruple H∗ v -groups." January 14, 2023. https://doi.org/10.5281/zenodo.7535977.

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It is well known that H&lowast; v -groups and groups are connected through regular relations. The purpose of this paper is to find a similar connection between neutrosophic H&lowast; v -groups and neutrosophic groups by using the concept of fundamental relations on Hv-groups. First, we characterize the complete parts of neutrosophic Hv-groups. Then we study their (strongly) regular relations. Finally, we find their fundamental group.

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21

Abdullah, Kargın, Dayan Azize, and Merve Şahin Necmiye. "Generalized Hamming Similarity Measure Based on Neutrosophic Quadruple Numbers and Its Applications to Law Sciences." February 18, 2021. https://doi.org/10.5281/zenodo.4549328.

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Neutrosophic quadruple numbers are the newest field studied in neutrosophy. Neutrosophic quadruple numbers, using the certain extent known data of an object or an idea, help us uncover their known part and moreover they allow us to evaluate the unknown part by the trueness, indeterminacy and falsity values. In this study, we generalized Hamming similarity measures for the generalized set-valued neutrosophic quadruple sets and numbers. We showed that generalized Hamming measure satisfies the similarity measure condition. Also, we generalized an algorithm for the generalized set-valued neutrosop

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22

NSS. "Operators Based On Multiple Generalized Set-Valued Neutrosophic Quadruple Sets." August 2, 2024. https://doi.org/10.5281/zenodo.13172844.

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 In this study, as a generalized form of multiple valued neutrosophic numbers and neutrosophic quadruple numbers, multiple generalized set-valued neutrosophic quadruple numbers and sets are defined and their basic properties are given. Thus, a new structure is obtained in which multi-valued neutrosophic sets and neutrosophic quadruple sets are used together. Also, optimistic union, pessimistic union, average union, pessimistic intersection, optimistic intersection, average intersection operators are defined for multiple generalized set-valued neutrosophic qua

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23

G., Muhiuddin, N. Al-Kenani Ahmad, Hwan Roh Eun, and Bae Jun Young. "Implicative Neutrosophic Quadruple BCK-Algebras and Ideals." May 20, 2019. https://doi.org/10.5281/zenodo.3044809.

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A neutrosophic set is initiated by Smarandache, and it is a novel tool to deal with vagueness considering the truth, indeterminacy and falsity memberships satisfying the condition that their sum is less than 3. The concept of neutrosophic quadruple numbers was introduced by Florentin Smarandache. Using this idea, Jun et al. introduced the notion of neutrosophic quadruple BCK/BCI-numbers, and studied neutrosophic quadruple BCK/BCI-algebras.

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24

Memet, Şahin, Kargın Abdullah, and Sena Uz Merve. "Generalized Euclid Measures Based on Generalized Set Valued Neutrosophic Quadruple Numbers and Multi Criteria Decision Making Applications." December 12, 2021. https://doi.org/10.5281/zenodo.5775188.

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In this article, we define firstly generalized Euclid distance measure and generalized Euclid similarity measure based on generalized set valued neutrosophic quadruple numbers. Also, we show that generalized Euclid distance measure and generalized Euclid similarity measure satisfy the distance measure conditions and similarity measure conditions, respectively. Furthermore, we define a score function for generalized Euclid similarity measure. In addition, we generalize an algorithm, for single valued neutrosophic set, based on generalized Euclid similarity measure and generalized set valued neu

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25

Sevilay, Şahin, Kargın Abdullah, and Yücel Murat. "Hausdorff Measures on Generalized Set Valued Neutrosophic Quadruple Numbers and Decision Making Applications for Adequacy of Online Education." February 18, 2021. https://doi.org/10.5281/zenodo.4549332.

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In this paper, we develop a new method of decision-making algorithm with Hausdorff distance and Hausdorff similarity measures based on generalized set-valued neutrosophic quadruple numbers. To establish the algorithm, we define Hausdorff distance measure and Hausdorff similarity measure on generalized setvalued neutrosophic quadruple. Next, we give a new method of decision-making application for impact of online learning on the learner. Also, we obtain different result from some previous applications (based on neutrosophic sets) for decision making algorithm. Thanks to our decision-making algo

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26

G., Muhiuddin, N. Al-Kenani Ahmad, Hwan Roh Eun, and Bae Jun Young. "Implicative Neutrosophic Quadruple BCK-Algebras and Ideals." May 20, 2019. https://doi.org/10.5281/zenodo.3045003.

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A neutrosophic set is initiated by Smarandache, and it is a novel tool to deal with vagueness considering the truth, indeterminacy and falsity memberships satisfying the condition that their sum is less than 3. The concept of neutrosophic quadruple numbers was introduced by Florentin Smarandache. Using

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27

Christianto, V., F. Smarandache, and M. Aslam. "How we can extend the standard deviation notion with neutrosophic interval and quadruple neutrosophic numbers." International Journal of Neutrosophic Science, 2020, 72–76. http://dx.doi.org/10.54216/ijns.020202.

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During scientific demonstrating of genuine specialized framework we can meet any sort and rate model vulnerability. Its reasons can be incognizance of modelers or information mistake. In this way, characterization of vulnerabilities, as for their sources, recognizes aleatory and epistemic ones. The aleatory vulnerability is an inalienable information variety related with the researched framework or its condition. Epistemic one is a vulnerability that is because of an absence of information on amounts or procedures of the framework or the earth [7]. Right now, we examine fourfold neutrosophic n

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28

V., Christianto, Smarandache F., and Aslam M. "How we can extend the standard deviation notion with neutrosophic interval and quadruple neutrosophic numbers." August 18, 2020. https://doi.org/10.5281/zenodo.3989247.

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During scientific demonstrating of genuine specialized framework we can meet any sort and rate model vulnerability. Its reasons can be incognizance of modelers or information mistake. In this way, characterization of vulnerabilities, as for their sources, recognizes aleatory and epistemic ones. The aleatory vulnerability is an inalienable information variety related with the researched framework or its condition. Epistemic one is a vulnerability that is because of an absence of information on amounts or procedures of the framework or the earth [7]. Right now, we examine fourfold neutrosophic n

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29

Okumus, Nuh, and Merve Sena Uz. "Decision Making Applications for Business Based on Generalized Set-Valued Neutrosophic Quadruple Sets." International Journal of Neutrosophic Science, 2022, 82–98. http://dx.doi.org/10.54216/ijns.180108.

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In this article, an algorithm is introduced that makes it easier for us to choose the right operator in order to overcome the possible problems faced by a business, small or huge. This algorithm is based on the Euclidean similarity measure on generalized set-valued neutrosophic quadruple numbers and sets. By using this algorithm, one can calculate the similarity of the criteria between the requested and already owned skills of the operators in general. By this means, one can make the right choice to hire the right operator without giving preferential treatment or personal favors. The closer th

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30

Sahin, Necmiye Merve, and Merve Sena Uz. "Multi-criteria Decision-making Applications Based on Set Valued Generalized Neutrosophic Quadruple Sets for Law." International Journal of Neutrosophic Science, 2021, 41–60. http://dx.doi.org/10.54216/ijns.170103.

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In this article, an algorithm has been introduced that enables judges to see the decisions that should be made in a way that is closest to the conscience and the law, without transferring the cases to the higher authorities, without anyone objecting to their decisions. This algorithm has been introduced depending on the generalized set-valued neutrosophic quadruple numbers and the Euclidean similarity measure in sets, what the decision is made by considering all the situations, regardless of which case the defendants come before the judge, how similar these decisions are to the legal decisions

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31

Florentin, Smarandache. "Introduction to the Symbolic Plithogenic Algebraic Structures (revisited)." January 14, 2023. https://doi.org/10.5281/zenodo.7536105.

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: In this paper, we recall and study the new type of algebraic structures called Symbolic Plithogenic Algebraic Structures. Their operations are given under the Absorbance Law and the Prevalence Order.

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