The significance of the contributions of congruences to the theory of connectednesses and disconnectednesses for topological spaces and graphs

TL;DR

This survey explores how congruences influence the theory of connectedness and disconnectedness in graphs and topological spaces, revealing new insights and unexpected differences in radical characterizations.

Contribution

It demonstrates that connectednesses and disconnectednesses can be characterized as Hoehnke radicals and uncovers surprising distinctions in radical properties within these categories.

Findings

Connectednesses characterized as Hoehnke radicals.

Existence of non-trivial connectednesses and disconnectednesses in loopless graphs.

Hoehnke radicals can degenerate in certain categories.

Abstract

This is a survey of some of the consequences of the recently introduced congruences on the theory of connectednesses (radical classes) and disconnectednesses (semisimple classes) of graphs and topological spaces. In particular, it is shown that the connectednesses and disconnectednesses can be obtained as Hoehnke radicals and a connectedness has a characterization in terms of congruences resembling the classical characterization of its algebraic counterpart using ideals for a radical class. But this approach has also shown that there are some unexpected differences and surprises: an ideal-hereditary Hoehnke radical of topological spaces or graphs need not be a Kurosh-Amitsur radical and in the category of graphs with no loops, non-trivial connectednesses and disconnectednesses exist, but all Hoehnke radicals degenerate.