Hereditary torsion theories for graphs (Q822610)

The paper revisits classical extensions of concepts from universal algebras to classes of graphs. The fundamentals of the `algebraization' of graphs are based on the concept of (strong) congruencies of graphs and their connection to graph homomorphisms and their kernels. A considerable part of the paper is devoted to the introduction of the considered concepts and a review of related results, such as, for example, the Isomorphism Theorems for graph homomorphisms.\par A \textit{Hoehnke radical} on a class of graphs is a function \( \rho \) assigning a congruence \( \rho_G \) to each graph \(G\) in the class, which `commutes' with homomorphisms from \(G\) into the members of the class, and assigns the identity congruence to the factor graph \( G/\rho_G \). In the context of graphs, ideal hereditary Hoehnke radicals are called \textit{hereditary torsion theories}, and radical and semisimple classes (consisting of graphs for which \( G/\rho_G \) is trivial, or \( \rho_G\) is the identity congruence, respectively) are called \textit{connectednesses} and \textit{disconnectednesses}. All ideal hereditary Hoehnke radicals of simple graphs with loops are classified, and as a consequence are shown to include radicals which are not Kurosh-Amitsur radicals.