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20 pages, 1424 KiB

Open AccessArticle

A Kind of Variation Symmetry: Tarski Associative Groupoids (TA-Groupoids) and Tarski Associative Neutrosophic Extended Triplet Groupoids (TA-NET-Groupoids)

by Xiaohong Zhang, Wangtao Yuan, Mingming Chen and Florentin Smarandache

Symmetry 2020, 12(5), 714; https://doi.org/10.3390/sym12050714 - 2 May 2020

Cited by 4 | Viewed by 2298

Abstract

The associative law reflects symmetry of operation, and other various variation associative laws reflect some generalized symmetries. In this paper, based on numerous literature and related topics such as function equation, non-associative groupoid and non-associative ring, we have introduced a new concept of [...] Read more.

The associative law reflects symmetry of operation, and other various variation associative laws reflect some generalized symmetries. In this paper, based on numerous literature and related topics such as function equation, non-associative groupoid and non-associative ring, we have introduced a new concept of Tarski associative groupoid (or transposition associative groupoid (TA-groupoid)), presented extensive examples, obtained basic properties and structural characteristics, and discussed the relationships among few non-associative groupoids. Moreover, we proposed a new concept of Tarski associative neutrosophic extended triplet groupoid (TA-NET-groupoid) and analyzed related properties. Finally, the following important result is proved: every TA-NET-groupoid is a disjoint union of some groups which are its subgroups. Full article

(This article belongs to the Special Issue New Development of Neutrosophic Probability, Neutrosophic Statistics, Neutrosophic Algebraic Structures, and Neutrosophic & Plithogenic Optimizations)

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21 pages, 454 KiB

Open AccessArticle

Regular CA-Groupoids and Cyclic Associative Neutrosophic Extended Triplet Groupoids (CA-NET-Groupoids) with Green Relations

by Wangtao Yuan and Xiaohong Zhang

Mathematics 2020, 8(2), 204; https://doi.org/10.3390/math8020204 - 6 Feb 2020

Cited by 6 | Viewed by 2235

Abstract

Based on the theories of AG-groupoid, neutrosophic extended triplet (NET) and semigroup, the characteristics of regular cyclic associative groupoids (CA-groupoids) and cyclic associative neutrosophic extended triplet groupoids (CA-NET-groupoids) are further studied, and some important results are obtained. In particular, the following conclusions are [...] Read more.

Based on the theories of AG-groupoid, neutrosophic extended triplet (NET) and semigroup, the characteristics of regular cyclic associative groupoids (CA-groupoids) and cyclic associative neutrosophic extended triplet groupoids (CA-NET-groupoids) are further studied, and some important results are obtained. In particular, the following conclusions are strictly proved: (1) an algebraic system is a regular CA-groupoid if and only if it is a CA-NET-groupoid; (2) if (S, *) is a regular CA-groupoid, then every element of S lies in a subgroup of S, and every

ℋ

-class in S is a group; and (3) an algebraic system is an inverse CA-groupoid if and only if it is a regular CA-groupoid and its idempotent elements are commutative. Moreover, the Green relations of CA-groupoids are investigated, and some examples are presented for studying the structure of regular CA-groupoids. Full article

(This article belongs to the Special Issue New Challenges in Neutrosophic Theory and Applications)

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13 pages, 744 KiB

Open AccessArticle

New Results on Neutrosophic Extended Triplet Groups Equipped with a Partial Order

by Xin Zhou, Ping Li, Florentin Smarandache and Ahmed Mostafa Khalil

Symmetry 2019, 11(12), 1514; https://doi.org/10.3390/sym11121514 - 13 Dec 2019

Cited by 8 | Viewed by 2710

Abstract

Neutrosophic extended triplet group (NETG) is a novel algebra structure and it is different from the classical group. The major concern of this paper is to present the concept of a partially ordered neutrosophic extended triplet group (po-NETG), which is a NETG equipped [...] Read more.

Neutrosophic extended triplet group (NETG) is a novel algebra structure and it is different from the classical group. The major concern of this paper is to present the concept of a partially ordered neutrosophic extended triplet group (po-NETG), which is a NETG equipped with a partial order that relates to its multiplicative operation, and consider properties and structure features of po-NETGs. Firstly, in a po-NETG, we propose the concepts of the positive cone and negative cone, and investigate the structure features of them. Secondly, we study the specificity of the positive cone in a partially ordered weak commutative neutrosophic extended triplet group (po-WCNETG). Finally, we introduce the concept of a po-NETG homomorphism between two po-NETGs, construct a po-NETG on a quotient set by providing a multiplication and a partial order, then we discuss some fundamental properties of them. Full article

(This article belongs to the Special Issue New Development of Neutrosophic Probability, Neutrosophic Statistics, Neutrosophic Algebraic Structures, and Neutrosophic & Plithogenic Optimizations)

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20 pages, 286 KiB

Open AccessArticle

Generalized Abel-Grassmann’s Neutrosophic Extended Triplet Loop

by Xiaogang An, Xiaohong Zhang and Yingcang Ma

Mathematics 2019, 7(12), 1206; https://doi.org/10.3390/math7121206 - 9 Dec 2019

Cited by 5 | Viewed by 1934

Abstract

A group is an algebraic system that characterizes symmetry. As a generalization of the concept of a group, semigroups and various non-associative groupoids can be considered as algebraic abstractions of generalized symmetry. In this paper, the notion of generalized Abel-Grassmann’s neutrosophic extended triplet [...] Read more.

A group is an algebraic system that characterizes symmetry. As a generalization of the concept of a group, semigroups and various non-associative groupoids can be considered as algebraic abstractions of generalized symmetry. In this paper, the notion of generalized Abel-Grassmann’s neutrosophic extended triplet loop (GAG-NET-Loop) is proposed and some properties are discussed. In particular, the following conclusions are strictly proved: (1) an algebraic system is an AG-NET-Loop if and only if it is a strong inverse AG-groupoid; (2) an algebraic system is a GAG-NET-Loop if and only if it is a quasi strong inverse AG-groupoid; (3) an algebraic system is a weak commutative GAG-NET-Loop if and only if it is a quasi Clifford AG-groupoid; and (4) a finite interlaced AG-(l,l)-Loop is a strong AG-(l,l)-Loop. Full article

(This article belongs to the Special Issue New Challenges in Neutrosophic Theory and Applications)

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15 pages, 806 KiB

Open AccessArticle

The Structure of Idempotents in Neutrosophic Rings and Neutrosophic Quadruple Rings

by Yingcang Ma, Xiaohong Zhang, Florentin Smarandache and Juanjuan Zhang

Symmetry 2019, 11(10), 1254; https://doi.org/10.3390/sym11101254 - 8 Oct 2019

Cited by 15 | Viewed by 2229

Abstract

This paper aims to reveal the structure of idempotents in neutrosophic rings and neutrosophic quadruple rings. First, all idempotents in neutrosophic rings

⟨ R \cup I ⟩

are given when R is

C, R, Q, Z

or

Z_{n}

. [...] Read more.

This paper aims to reveal the structure of idempotents in neutrosophic rings and neutrosophic quadruple rings. First, all idempotents in neutrosophic rings

⟨ R \cup I ⟩

are given when R is

C, R, Q, Z

or

Z_{n}

. Secondly, the neutrosophic quadruple ring

⟨ R \cup T \cup I \cup F ⟩

is introduced and all idempotents in neutrosophic quadruple rings

⟨ C \cup T \cup I \cup F ⟩, ⟨ R \cup T \cup I \cup F ⟩,

⟨ Q \cup T \cup I \cup F ⟩, ⟨ Z \cup T \cup I \cup F ⟩

and

⟨ Z_{n} \cup T \cup I \cup F ⟩

are also given. Furthermore, the algorithms for solving the idempotents in

⟨ Z_{n} \cup I ⟩

and

⟨ Z_{n} \cup T \cup I \cup F ⟩

for each nonnegative integer n are provided. Lastly, as a general result, if all idempotents in any ring R are known, then the structure of idempotents in neutrosophic ring

⟨ R \cup I ⟩

and neutrosophic quadruple ring

⟨ R \cup T \cup I \cup F ⟩

can be determined. Full article

(This article belongs to the Special Issue New types of Neutrosophic Set/Logic/Probability, Neutrosophic Over-/Under-/Off-Set, Neutrosophic Refined Set, and their Extension to Plithogenic Set/Logic/Probability, with Applications)

18 pages, 2910 KiB

Open AccessArticle

Symmetry in Hyperstructure: Neutrosophic Extended Triplet Semihypergroups and Regular Hypergroups

by Xiaohong Zhang, Florentin Smarandache and Yingcang Ma

Symmetry 2019, 11(10), 1217; https://doi.org/10.3390/sym11101217 - 1 Oct 2019

Cited by 5 | Viewed by 2112

Abstract

The symmetry of hyperoperation is expressed by hypergroup, more extensive hyperalgebraic structures than hypergroups are studied in this paper. The new concepts of neutrosophic extended triplet semihypergroup (NET- semihypergroup) and neutrosophic extended triplet hypergroup (NET-hypergroup) are firstly introduced, some basic properties are obtained, [...] Read more.

The symmetry of hyperoperation is expressed by hypergroup, more extensive hyperalgebraic structures than hypergroups are studied in this paper. The new concepts of neutrosophic extended triplet semihypergroup (NET- semihypergroup) and neutrosophic extended triplet hypergroup (NET-hypergroup) are firstly introduced, some basic properties are obtained, and the relationships among NET- semihypergroups, regular semihypergroups, NET-hypergroups and regular hypergroups are systematically are investigated. Moreover, pure NET-semihypergroup and pure NET-hypergroup are investigated, and a strucuture theorem of commutative pure NET-semihypergroup is established. Finally, a new notion of weak commutative NET-semihypergroup is proposed, some important examples are obtained by software MATLAB, and the following important result is proved: every pure and weak commutative NET-semihypergroup is a disjoint union of some regular hypergroups which are its subhypergroups. Full article

(This article belongs to the Special Issue New types of Neutrosophic Set/Logic/Probability, Neutrosophic Over-/Under-/Off-Set, Neutrosophic Refined Set, and their Extension to Plithogenic Set/Logic/Probability, with Applications)

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13 pages, 300 KiB

Open AccessArticle

Study on the Algebraic Structure of Refined Neutrosophic Numbers

by Qiaoyan Li, Yingcang Ma, Xiaohong Zhang and Juanjuan Zhang

Symmetry 2019, 11(8), 954; https://doi.org/10.3390/sym11080954 - 27 Jul 2019

Cited by 1 | Viewed by 2467

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue New types of Neutrosophic Set/Logic/Probability, Neutrosophic Over-/Under-/Off-Set, Neutrosophic Refined Set, and their Extension to Plithogenic Set/Logic/Probability, with Applications)

15 pages, 462 KiB

Open AccessArticle

Neutrosophic Extended Triplet Group Based on Neutrosophic Quadruple Numbers

by Qiaoyan Li, Yingcang Ma, Xiaohong Zhang and Juanjuan Zhang

Symmetry 2019, 11(5), 696; https://doi.org/10.3390/sym11050696 - 21 May 2019

Cited by 10 | Viewed by 2712

Abstract

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 [...] Read more.

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 respect to the multiplication operator; (4) the union of the set which has same neutral element is a partition of four-dimensional space. Full article

(This article belongs to the Special Issue New types of Neutrosophic Set/Logic/Probability, Neutrosophic Over-/Under-/Off-Set, Neutrosophic Refined Set, and their Extension to Plithogenic Set/Logic/Probability, with Applications)

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13 pages, 1204 KiB

Open AccessArticle

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

by Xiaoying Wu and Xiaohong Zhang

Mathematics 2019, 7(3), 268; https://doi.org/10.3390/math7030268 - 15 Mar 2019

Cited by 22 | Viewed by 2621

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue General Algebraic Structures)

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16 pages, 256 KiB

Open AccessArticle

Generalized Neutrosophic Extended Triplet Group

by Yingcang Ma, Xiaohong Zhang, Xiaofei Yang and Xin Zhou

Symmetry 2019, 11(3), 327; https://doi.org/10.3390/sym11030327 - 5 Mar 2019

Cited by 30 | Viewed by 3086

Abstract

Neutrosophic extended triplet group is a new algebra structure and is different from the classical group. In this paper, the notion of generalized neutrosophic extended triplet group is proposed and some properties are discussed. In particular, the following conclusions are strictly proved: (1) [...] Read more.

Neutrosophic extended triplet group is a new algebra structure and is different from the classical group. In this paper, the notion of generalized neutrosophic extended triplet group is proposed and some properties are discussed. In particular, the following conclusions are strictly proved: (1) an algebraic system is a generalized neutrosophic extended triplet group if and only if it is a quasi-completely regular semigroup; (2) an algebraic system is a weak commutative generalized neutrosophic extended triplet group if and only if it is a quasi-Clifford semigroup; (3) for each

n \in Z^{+}, n \geq 2, (Z_{n}, \otimes)

is a commutative generalized neutrosophic extended triplet group; (4) for each

n \in Z^{+}, n \geq 2, (Z_{n}, \otimes)

is a commutative neutrosophic extended triplet group if and only if

n = p_{1} p_{2} \dots p_{m}

, i.e., the factorization of n has only single factor. Full article

(This article belongs to the Special Issue New types of Neutrosophic Set/Logic/Probability, Neutrosophic Over-/Under-/Off-Set, Neutrosophic Refined Set, and their Extension to Plithogenic Set/Logic/Probability, with Applications)

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12 pages, 230 KiB

Open AccessArticle

On Homomorphism Theorem for Perfect Neutrosophic Extended Triplet Groups

by Xiaohong Zhang, Xiaoyan Mao, Florentin Smarandache and Choonkil Park

Information 2018, 9(9), 237; https://doi.org/10.3390/info9090237 - 18 Sep 2018

Cited by 5 | Viewed by 3056

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Section Artificial Intelligence)

14 pages, 786 KiB

Open AccessArticle

Fundamental Homomorphism Theorems for Neutrosophic Extended Triplet Groups

by Mehmet Çelik, Moges Mekonnen Shalla and Necati Olgun

Symmetry 2018, 10(8), 321; https://doi.org/10.3390/sym10080321 - 3 Aug 2018

Cited by 9 | Viewed by 3782

Abstract

In classical group theory, homomorphism and isomorphism are significant to study the relation between two algebraic systems. Through this article, we propose neutro-homomorphism and neutro-isomorphism for the neutrosophic extended triplet group (NETG) which plays a significant role in the theory of neutrosophic triplet [...] Read more.

In classical group theory, homomorphism and isomorphism are significant to study the relation between two algebraic systems. Through this article, we propose neutro-homomorphism and neutro-isomorphism for the neutrosophic extended triplet group (NETG) which plays a significant role in the theory of neutrosophic triplet algebraic structures. Then, we define neutro-monomorphism, neutro-epimorphism, and neutro-automorphism. We give and prove some theorems related to these structures. Furthermore, the Fundamental homomorphism theorem for the NETG is given and some special cases are discussed. First and second neutro-isomorphism theorems are stated. Finally, by applying homomorphism theorems to neutrosophic extended triplet algebraic structures, we have examined how closely different systems are related. Full article

(This article belongs to the Special Issue Algebraic Structures of Neutrosophic Triplets, Neutrosophic Duplets, or Neutrosophic Multisets)

13 pages, 2711 KiB

Open AccessArticle

Neutrosophic Triplet Cosets and Quotient Groups

by Mikail Bal, Moges Mekonnen Shalla and Necati Olgun

Symmetry 2018, 10(4), 126; https://doi.org/10.3390/sym10040126 - 20 Apr 2018

Cited by 29 | Viewed by 5020

Abstract

In this paper, by utilizing the concept of a neutrosophic extended triplet (NET), we define the neutrosophic image, neutrosophic inverse-image, neutrosophic kernel, and the NET subgroup. The notion of the neutrosophic triplet coset and its relation with the classical coset are defined and [...] Read more.

In this paper, by utilizing the concept of a neutrosophic extended triplet (NET), we define the neutrosophic image, neutrosophic inverse-image, neutrosophic kernel, and the NET subgroup. The notion of the neutrosophic triplet coset and its relation with the classical coset are defined and the properties of the neutrosophic triplet cosets are given. Furthermore, the neutrosophic triplet normal subgroups, and neutrosophic triplet quotient groups are studied. Full article

(This article belongs to the Special Issue Algebraic Structures of Neutrosophic Triplets, Neutrosophic Duplets, or Neutrosophic Multisets)

11 pages, 235 KiB

Open AccessArticle

Study on the Development of Neutrosophic Triplet Ring and Neutrosophic Triplet Field

by Mumtaz Ali, Florentin Smarandache and Mohsin Khan

Mathematics 2018, 6(4), 46; https://doi.org/10.3390/math6040046 - 23 Mar 2018

Cited by 23 | Viewed by 3250 | Correction

Abstract

Rings and fields are significant algebraic structures in algebra and both of them are based on the group structure. In this paper, we attempt to extend the notion of a neutrosophic triplet group to a neutrosophic triplet ring and a neutrosophic triplet field. [...] Read more.

Rings and fields are significant algebraic structures in algebra and both of them are based on the group structure. In this paper, we attempt to extend the notion of a neutrosophic triplet group to a neutrosophic triplet ring and a neutrosophic triplet field. We introduce a neutrosophic triplet ring and study some of its basic properties. Further, we define the zero divisor, neutrosophic triplet subring, neutrosophic triplet ideal, nilpotent integral neutrosophic triplet domain, and neutrosophic triplet ring homomorphism. Finally, we introduce a neutrosophic triplet field. Full article

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