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The concept of unity roots plays a central role in the theory of field extensions and polynomials roots' computing. The objective of this paper is to find the algebraic formula for computing the symbolic 2-plithogenic and 3-plithogenic complex roots of unity, where a general formula will be provided with many related examples up to the exponent 3. [ABSTRACT FROM AUTHOR]

The objective of this paper is to study for the first time the concept of square real matrices with symbolic 2-plithogenic split-complex entries. Many of their algebraic properties will be discussed and handled, where we find the formula of computing inverses, exponents, and powers of these matrices by building a ring isomorphism between the ring of split-complex symbolic 2-plithogenic matrices and the direct product of the symbolic 2-plithogenic matrices with itself. Also, we give the interested reader many related examples to clarify the validity of our work. [ABSTRACT FROM AUTHOR]

The goal of this paper is to define for the first time the concept of symbolic 2-plithogenic weak fuzzy complex number as new generalization generated by combining real numbers with symbolic 2-plithogenic numbers. We study the elementary properties of this new class such as Invertibility and nilpotency, with many related examples that explain its novelty. [ABSTRACT FROM AUTHOR]

Complex numbers have been studied in previous papers, but the papers did not deal with the conversion from algebraic form of a neutrosophic complex number form to exponential or trigonometric form and vice versa, and this prompted us to search for a method that facilitates this conversion process, as this paper dealt with how to move from algebraic form of a neutrosophic complex number to exponential or trigonometric form and vice versa, also, we discussed the roots from order n of a neutrosophic complex number, whether this number is given in algebraic or trigonometric form. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the neutrosophic complex finite rings C(Zn) and C(< Zn ∪ I >), and to give a classification theorem of these rings. Also, this work introduces full solutions for 12 Kandasamy-Smarandache open problems concerning these structures of generalized rings modulo integers. Also, a necessary and sufficient condition of invertibility in C(Zn) and C(< Zn ∪ I >) is presented as a partial solution of the famous group of units problem. [ABSTRACT FROM AUTHOR]

In this paper, we make a short history about: the neutrosophic set, neutrosophic numerical components and neutrosophic literal components, neutrosophic numbers, neutrosophic intervals, neutrosophic hypercomplex numbers of dimension n, and elementary neutrosophic algebraic structures. Afterwards, their generalizations to refined neutrosophic set, respectively refined neutrosophic numerical and literal components, then refined neutrosophic numbers and refined neutrosophic algebraic structures. The aim of this paper is to construct examples of splitting the literal indeterminacy (I) into literal sub-indeterminacies (I1, I2, …, Ir), and to define a multiplication law of these literal sub-indeterminacies in order to be able to build refined I-neutrosophic algebraic structures. Also, examples of splitting the numerical indeterminacy (i) into numerical sub-indeterminacies, and examples of splitting neutrosophic numerical components into neutrosophic numerical sub-components are given. [ABSTRACT FROM AUTHOR]

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 this way, we define new structures using the together set valued neutrosophic quadruple number and neutrosophic triplet group. Thus, we obtain new results for set valued neutrosophic quadruple numbers and neutrosophic triplet groups based on set valued neutrosophic quadruple number. [ABSTRACT FROM AUTHOR]

The interval complex single-valued neutrosophic hypersoft set (Ξ-set), together with its features and set-theoretic operations, is a new mathematical structure that is discussed in this article. For managing ambiguous and uncertain knowledge, the suggested structure integrates the interval complex single-valued neutrosophic set and hypersoft set. These two elements have already been regarded as trustworthy settings. The first component has the ability to manage information on interval and periodic types, while the second offers a multi-argument domain for concurrent consideration of numerous sub-attributes. The Ξ-set is used to aggregate these sets, allowing for the fusion of various qualities and any related uncertainty. The resultant aggregated sets, which take into account both the attribute values and the associated uncertainty, give a thorough representation of the decision aspect. To assist in decision-making, the method calculates how similar several options are to the optimum option using a distance-based similarity metric. By contrasting the combined sets of several options, the system determines the best option based on the specified selection criteria. Decision-makers can evaluate how changing attribute values may influence their choices using the suggested strategy’s endorsement of sensitivity analysis. The efficacy of the recommended decision-support mechanism is demonstrated through a case study with a real-world choice dilemma. The results show how well the framework can handle ambiguity and uncertainty while providing decision-makers with meaningful insights and encouraging rational choices. Finally, the multi-attribute decision-support system based on aggregations of Ξ-set provides a reliable framework for dealing with difficult choice issues that are characterized by ambiguity and vagueness. [ABSTRACT FROM AUTHOR]