NeutroAlgebras consist of algebras containing at least one NeutroAxiom. In the literature, there is a method for decision-making in a migrational situation that uses a NeutroAlgebra generated by the combining function in Prospector. This method is an important hit in showing both the utility and theoretical significance of NeutroAlgebras since it uses this abstract tool for solving a real-life problem. To the authors' knowledge, this is the only approach to NeutroAlgebras which is not exclusively theoretical. This chapter presents the rigorous mathematical proofs of the properties of that NeutroAlgebra because so far the theoretical approaches to the proposed method are minimal. The deeper mathematical knowledge of this specific NeutroAlgebra is a contribution to the general theory on this subject. The authors expect that this chapter will serve as dissemination of NeutroAlgebras among undergraduate and graduate students who may be interested in this issue.
TopIntroduction
The theory of NeutroAlgebras, which has recently been introduced by F. Smarandache, generalizes the classical theory of Algebras and Partial Algebras within the framework of Neutrosophy (Smarandache, 2020a, 2020b). NeutroAlgebras preserve the study of algebraic structures defined as ordered pairs formed by a set of elements and algebraic operations. The main difference between the NeutroAlgebras concerning algebras and partial algebras is that the former ones contain at least one NeutroAxiom which is an axiom where there are two types of elements, those that comply with the axiom and those that do not comply with it.
Following the main idea of Neutrosophy (Smarandache, 2005), given an Algebra (Axiom) <A>, there is a triad (<A>, <NeutA>, <AntiA>) where the Algebra (Axiom) <A> is 100% true or true for all the elements, NeutroAlgebras (NeutroAxioms) <NeutA> are satisfied only by a part of the elements, while AntiAlgebras (AntiAxioms) <AntiA> are not satisfied by any of the elements of the set, (Smarandache, 2020a).
This new approach to one of the most classical branches of mathematics constitutes a challenge for the understanding of these novel ideas. Classical algebra is based on mathematical logic where only axioms 100% true are permitted. From a theoretical point of view, there may be doubts about the usefulness of modeling indeterminacy and uncertainty in a well-established and solid mathematical branch such as algebra.
The other controversial aspect is about the usefulness of studying NeutroAlgebras due to a large number of applications of classical algebras in other sciences, such as Physics, Chemistry, Engineering, among others. All these sciences need knowledge on the theory of groups, Galois theory, theory of linear spaces, among other sub-branches of the classical algebras. Thus, some scientists could argue that these classical theories are sufficient to model scientific and real-life problems, and NeutroAlgebras can be considered unnecessary, (Isaacs, 2008).
In the literature review on NeutroAlgebras carried out by the authors, some examples of NeutroGroups, NeutroRings, among others, were found, which essentially only constitute theoretical approaches to these subjects, (Abobala, 2021; Agboola, 2020a; Gayen et al., 2020; Ibrahim & Agboola, 2020; Sankari & Abobala, 2020). The only exception of one NeutroAlgebra with practical application is the one based on the uninorm defined from the combining function in the Prospector expert system. This consists in a decision-making method that was designed for modeling the study of migration, (Silva-Jiménez, Valenzuela-Mayorga, Roja-Ubilla, & Batista-Hernández, 2021).
Uninorms are operators defined for the first time in the framework of fuzzy logic, which generalizes the axiomatic of t-norm, representing the logical AND, and t-conorm, representing the logical OR (Fodor, Yager, & Rybalov, 1997). In addition to the axioms of commutativity, associativity, non-decreasing, in uninorms, the axiom of the existence of a neutral element is extended to a value other than 0 and 1. This type of operator has been applied in countless problems of many kinds, (Bordignon & Gomide, 2014; González-Hidalgo, Massanet, Mir, & Ruiz-Aguilera, 2014; Yager, 2001; Yager & Kreinovich, 2003; Yager & Rybalov, 1996). It has also been generalized in other frameworks such as the theories of intuitionistic fuzzy sets (Deschrijver & Kerre, 2004), neutrosophic sets (González-Caballero, Leyva-Vázquez, & Smarandache, 2021), and neutrosophic offsets (González-Caballero, Smarandache, & Leyva-Vázquez, 2019).
On the other hand, the combining function in Prospector has transcended the framework of this expert system and is itself a subject of study (De-Baets & Fodor, 1999). Prospector expert system is dedicated to mineral search prediction (Hart, Duda, & Einaudi, 1978; Marík & Kouba, 1991; Zhang & Luo, 1999).
The application of a NeutroAlgebra to solve a real-life problem in (Silva-Jiménez et al., 2021) served to ratify the idea presented by Smarandache that NeutroAlgebras are important for modeling in the so-called Soft Sciences, where the existence of 100% true axioms is not the generality. That is why from the pedagogical point of view these NeutroAlgebras are a good starting point to teach the general theory of NeutroAlgebras to students of careers where mathematics are essentials, and which there is an important social component, for example, in economics and political sciences.