สมบัติเชิงโครงสร้างของเซตย่อยและฟังก์ชัน 𝑩-ปฏิสาทิสสัณฐานบน 𝑩 -พีชคณิต:

Poonchayar Patthanangkoor; Kanittakorn  Moonchaisook; Kittiporn  Sapjarean; Niyom  Kanpanit

Authors

Poonchayar Patthanangkoor Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, Rangsit Centre, Thailand
Kanittakorn Moonchaisook Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, Rangsit Centre, Thailand
Kittiporn Sapjarean Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, Rangsit Centre, Thailand
Niyom Kanpanit Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, Rangsit Centre, Thailand

Keywords:

𝘉-algebra, ideal, 𝘉-antihomomorphism

Abstract

Background and Objectives: Algebraic structures of non-empty sets equipped with binary operations represent a fundamental subject in abstract algebra that has been extensively studied. This research focuses on an intriguing algebraic structure known as B-algebras. Although various properties of B-algebras have been continuously investigated, in-depth details regarding the properties of certain specific subsets and the construction of algebraic structures on functional sets have not yet been comprehensively explored. The present research aims to address two primary objectives. First, it investigates and proves the structural properties of the subsets $M(X)$ and $C(X)$ for any $x\in X$ , where $X$ is a B-algebra, while identifying the necessary and sufficient conditions for the subset $M(X)$ to qualify as an ideal and a B-subalgebra. This investigation provides a deeper insight into internal structures that have remained unexplored in previous research. Second, the study examines B-antihomomorphisms to establish an algebraic structure on the set $AHom(X,Y)$ , which consists of all B-antihomomorphisms from $X$ to $Y$ , where $X$ and $Y$ are B-algebras, under a newly defined binary operation. The findings reveal that, under suitable conditions, $AHom(X,Y)$ constitutes a commutative B-algebra with associative properties. Consequently, this study extends the theoretical framework and establishes new algebraic properties, contributing to a more comprehensive understanding of B-algebra theory.

Methodology: This research relies on the definitions and theorems of B-algebras to investigate various algebraic properties. The methodology is divided into two main stages: the study of the structure and properties of B-algebra subsets, and the study of B-antihomomorphism functions. The first stage focused on exploring the properties of two specific sub-structures: $M(X)$ and $C(X)$ , which are defined for an arbitrary element $x$ of the B-algebra . The sets $M(X)$ and $C(X)$ are defined as $M(X)=\left\{a\in X(x^{*}a)^{^{*}}x=a\forall x\in X\right\}$ and $C(X)=\left\{y\in X(x^{*}y)^{^{*}}x=x\right\}$ $C(X)=\left\{y\in X(x^{*}y)^{^{*}}x=x\right\}$ , respectively. The study established the necessary conditions under which $M(X)$ constitutes both an ideal and a B-subalgebra of $X$ The demonstration utilized the fundamental B-algebra axioms to verify the closure and structural properties required by the definitions of a B-subalgebra and an ideal. The second stage of the study focused on investigating the properties of B-antihomomorphisms and characterizing the resulting structure of the set $AHom(X,Y)$ , defined as the collection of all such mappings from a B-algebra $X$ to a B-algebra $Y$ . The core of the methodology here was to define a binary operation on $AHom(X,Y)$ and use the established properties of B-algebras $X$ and $Y$ (specifically the constraints that $Y$ is a commutative and associative B-algebra) to prove that $AHom(X,Y)$ satisfies all the axioms of a B-algebra.

Main Results: This research established additional properties of B-algebras beyond those found in previous studies concerning B-algebras. It was found that in the case where $X$ is a commutative B-algebra, the subset $M(X)$ is both an ideal and a B-subalgebra of $X$ . Furthermore, it was found that the subset $A=\left\{0,a\right\}$ is an ideal of $X$ and a B-subalgebra of $X$ for all $a\in X$ . In addition, it was found that $y\in C(x)$ if and only if $x\in C(y)$ for all $x,y\in X$ . Also, we found that $x\in M(X)$ if and only if $x\in C(x)$ , and if $x\in C(y)$ and $y\in C(z)$ then $x=z$ , where $x,y,z\in X$ . Let $X$ be an arbitrary B-algebra, and let $Y$ be a B-algebra that satisfies the additional properties of commutativity and associativity with $MY=Y$ . We see that the zero function is a B-antihomomorphism, the set of all B-antihomomorphisms from $X$ to $Y$ , denoted $AHom(X,Y)$ , is non-empty. By defining a suitable binary operation on this set, it was successfully demonstrated that $AHom(X,Y)$ , when equipped with this induced operation and the zero function, satisfies all the defining axioms of a B-algebra. Specifically $AHom(X,Y)$ , is not merely a B-algebra but is also a commutative B-algebra and possesses the associative property.

Conclusions: The study found that B-algebras possess several interesting structures and properties. Consequently, numerous important properties related to B-algebras were successfully proven. It was definitively proven that the subset $M(X)$ ( the set of all medial elements of $X$ ) in a commutative B-algebra assumes the dual role of both an ideal and a B-subalgebra. Furthermore, the properties of the subset $C(x)$ were provided. Crucially, the study successfully proved the existence of an algebraic structure on the set of B-antihomomorphismsm maps. It was concluded that the set of all B-antihomomorphisms, $AHom(X,Y)$ , itself forms a commutative and associative B-algebra, provided the codomain $Y$ satisfies these same conditions with $M(Y)=Y$ . This establishes a new class of B-algebras derived from the structure of B-antihomomorphism functions. These research findings contribute to a deeper understanding of B-algebras and open up new avenues for future investigation, particularly in the study of ideals, quotient algebras, and derivations defined on the newly established B-algebra $AHom(X,Y)$ .

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