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

Abstract

This is a survey of some of the consequences of the recently introduced congruences on the theory of connectednesses (radical classes)  and disconnected-nesses (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.