An Investigation of Ring Structure of Graph Magma Algebras

Abstract

Let $G=(V,E)$ be a simple directed graph, where $V$ is any set. By "simple directed graph" means that there exists at most one edge from $u$ to $v$ for $u,v\in V$ (i.e, a directed graph can have loops but not multiple edges.) Consider, in addition, a symbol $0\in V$ and an operation on $S=V\cup \{0\}$ via the rule: $uv=u$ if $u,v\in E$ and $uv=0$, otherwise. The element $0$ is called the annihilator element of $S$. This structure is called \textit{graph magma induced by $G$} and is denoted by $M(G)$. $R=A[G]$ is a \textit{graph magma algebra} if it has $\mathcal{B}=V\cup \{1\}$ as a basis and, for $u,v\in V$, $uv=u$ if $(u,v)\in E$ and $uv=0$, otherwise. The aim of this thesis is to investigate the ring structure of graph magma algebras generated by associative graphs and certain special ideals of these rings within the framework of the Diaz-Boils and Lopez-Permouth study.This thesis consists of four chapters. In the first part of our thesis, we give a survey of the literature on graph magma algebras. In the second chapter, we offer fundamental background information on definitions and theorems in ring theory, module theory, and graph theory. In the third part of the thesis, we investigate graph magma algebras. It will be shown how a graph magma algebra is constructed, and the problem of when two associative graphs induce an isomorphic algebra will be characterized. In the fourth chapter, we first determined the Jacobson radical and analyzed simple left and right modules with a single vertex for graph magma algebras induced by graphs with infinitely many non-null connected components. We examined characterization graph magma algebras with finitely many non-null connected components. These rings were identified as semiperfect rings and we examined the conditions under which semiperfect algebras can arise as graph magma algebras. Furthermore, we investigate the right and left socle and the singular ideal of graph magma algebras with infinitely many non-null connected components. Lastly, we studied commutative graph magma algebras with infinitely many non-null connected components and examined the characterization of those with finitely many non-null connected components. In the last section, we examined algebras with bases formed by the vertices of the components $N_1\oplus K_1$ and $N_p\oplus K_1$, for this focused on upper-triangular and lower-triangular matrix algebras.