On relatively elementary definability of graph classes in the class of semigroups

Abstract

Based on the previously obtained concrete characterization of the endomorphism semigroups of quasi-acyclic reflexive graphs we prove the relatively elementary definability of the class of such graphs in the class of all semigroups. It will permit us to investigate for such graphs the abstract representation problem for the endomorphism semigroups of graphs and the problem of elementary definability of graphs by their endomorphism semigroups.