On Neutrosophic Quadruple Groups

As generalizations and alternatives of classical algebraic structures there have been introduced in 2019 the NeutroAlgebraic structures (or NeutroAlgebras) and AntiAlgebraic structures (or AntiAlgebras). Unlike the classical algebraic structures, where all operations are well defined and all axioms are totally true, in NeutroAlgebras and AntiAlgebras, the operations may be partially well defined and the axioms partially true or, respectively, totally outer-defined and the axioms totally false. These NeutroAlgebras and AntiAlgebras form a new field of research, which is inspired from our real world. In this paper, we study neutrosophic quadruple algebraic structures and NeutroQuadrupleAlgebraicStructures. NeutroQuadrupleGroup is studied in particular and several examples are provided. It is shown that (NQ(Z),÷)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(NQ({\mathbb {Z}}),\div )$$\end{document} is a NeutroQuadrupleGroup. Substructures of NeutroQuadrupleGroups are also presented with examples.


Introduction
It was started from Paradoxism, then to Neutrosophy, and afterwards to Neutrosophic Set and Neutrosophic Algebraic Structures. Paradoxism [21] is an international movement in science and culture, founded by Smarandache in 1980 s, based on excessive use of antitheses, oxymoron, contradictions, and paradoxes. During the 3 decades (1980-2020), hundreds of authors from tens of countries around the globe contributed papers to 15 international paradoxist anthologies. In 1995, Smarandache extended the paradoxism (based on opposites) to a new branch of philosophy called neutrosophy (based on opposites and their neutrals) that gave birth to many scientific branches, such as neutrosophic logic, neutrosophic set, neutrosophic probability and statistics, neutrosophic algebraic structures, and so on with multiple applications in engineering, computer science, administrative work, medical research etc. Neutrosophy is an extension of Yin-Yang Ancient Chinese Philosophy and of course of Dialectics. From Classical Algebraic Structures to NeutroAlgebraic Structures and AntiAlgebraicStructures. In 2019 and 2020, Smarandache [16][17][18] generalized the classical Algebraic Structures to NeutroAlgebraicStructures (or NeutroAlgebras) whose operations and axioms are partially true, partially indeterminate, and partially false as extensions of Partial Algebra, and to AntiAlgebraic Structures (or AntiAlgebra) whose operations and axioms are totally false. By considering a space and an operation defined on, in general, it does not mean that the operation is well defined for all elements of the space. We have three cases, as in neutrosophy: either the operation is well defined (as in classical algebraic structures), or partially defined and partially undefined, or partially outer-defined. Similarly, in general by defining an axiom on a given space under some given operations it does not mean that the axion is true for all elements of the space. Again we gave three cases as in neutrosophy: the axiom is true for all elements (as in classical algebraic structures), or the axiom is partially true and partially false, or the axiom is false for all elements. Motivation is the fact that in mathematics, in general, by defining an operation on a given set it does not mean that the operation is automatically well defined, but many times it is only partially well defined. Similarly, by defining an axiom on a given set, in general it does not mean that the axiom is true for all elements, but only partially true (i.e. true for some elements and maybe false for other elements). In the present paper, we study neutrosophic quadruple algebraic structures and NeutroQuadru-pleAlgebraicStructures. NeutroQuadrupleGroup is studied in particular and several examples are provided. It is shown that (NQ(ℤ), ÷) is a NeutroQuadrupleGroup. Substructures of NeutroQuadrupleGroups are also presented with examples.

Operation, NeutroOperation, AntiOperation
When we define an operation on a given set, it does not automatically mean that the operation is well defined. There are three possibilities: • The operation is well-defined (or inner-defined) for all set's elements (as in classical algebraic structures this is classical Operation). • The operation if well-defined for some elements, indeterminate for other elements, and outer-defined for others elements (this is NeutroOperation). • The operation is outer-defined for all set's elements (this is AntiOperation).

Axiom, NeutroAxiom, AntiAxiom
Similarly for an axiom, defined on a given set, endowed with some operation(s). When we define an axiom on a given set, it does not automatically mean that the axiom is true for all set's elements. We have three possibilities again: • The axiom is true for all set's elements (totally true) (as in classical algebraic structures; this is a classical Axiom). • The axiom if true for some elements, indeterminate for other elements, and false for other elements (this is Neu-troAxiom). • The axiom is false for all set's elements (this is AntiAxiom).

Algebra, NeutroAlgebra, AntiAlgebra
• An algebraic structure whose all operations are welldefined and all axioms are totally true is called Classical Algebraic Structure (or Algebra). • An algebraic structure that has at least one NeutroOperation or one NeutroAxiom (and no AntiOperation and no AntiAxiom) is called NeutroAlgebraic Structure (or NeutroAlgebra). • An algebraic structure that has at least one AntiOperation or Anti Axiom is called AntiAlgebraic Structure (or AntiAlgebra).
and degree of nonconfidence (F). N is a four-dimensional vector that can also be written as: N = (a, b, c, d).
There are transcendental, irrational, etc. numbers that are not well known, they are only partially known and partially unknown, they may have infinitely many decimals. Not even the most modern supercomputers can compute more than a few thousands decimals, but the infinitely many left decimals still remain unknown. Therefore, such numbers are very little known (because only a finite number of decimals are known), and infinitely unknown (because an infinite number of decimals are unknown). Take for example:

Definition 1 A neutrosophic set of quadruple numbers denoted by NQ(X) is a set defined by
where T, I, F have their usual neutrosophic logic meanings.
Definition 2 A neutrosophic quadruple number is a number of the form (a, bT, cI, dF) ∈ NQ(X) . For any neutrosophic quadruple number (a, bT, cI, dF) representing any entity which may be a number, an idea, an object, etc., a is called the known part and (bT, cI, dF) is called the unknown part. Two neutrosophic quadruple numbers x = (a, bT, cI, dF) and y = (e, fT, gI, hF) are said to be equal written x = y if and , NQ(ℤ) and NQ(ℂ) are neutrosophic sets of quadruple natural, integers, rationals, real and complex numbers respectively.

Example 2 The following
are examples of neutrosophic quadruple of integers, real and complex numbers, respectively.

D e f i n i t i o n
. We define the following: Definition 4 Let a = (a 1 , a 2 T, a 3 I, a 4 F) ∈ NQ(X) and let be any scalar which may be real or complex, the scalar product .a is defined by  x  Then Example 4 From Example 2, we obtain the following: (i) For the prevalence order T ≻ I ≻ F , we have (ii) For the prevalence order T ≺ I ≺ F , we have  = (a 1 , a 2 T, a 3 I, a 4 F).(b 1 , b 2 T, b 3 I, b 4 By similarly assuming the prevalence order T ≺ I ≺ F , we obtain from Eq. (1) a system of linear equations in unknowns x, y, z and w.

Neutrosophic Quadruple Algebraic Structures and Neutrosophic Quadruple Algebraic Hyper-structures
Let NQ(X) be a neutrosophic quadruple set and let * ∶ NQ(X) × NQ(X) → NQ(X) be a classical binary operation on NQ(X). The couple (NQ(X), * ) is called a neutrosophic quadruple algebraic structure. The structure (NQ(X), * ) is named according to the classical laws and axioms satisfied or obeyed by * . If * ∶ NQ(X) × NQ(X) → ℙ(NQ(X)) is the classical hyper operation on NQ(X). Then the couple (NQ(X), * ) is called a neutrosophic quadruple hyper-algebraic structure; and the hyper-structure (NQ(X), * ) is named according to the classical laws and axioms satisfied by * .