Residual properties of pre-bipartite digraphs

Abstract

We say that a directed graph is pre-bipartite if its symmetric closure is bipartite. We will show that the class \({\mathcal{B}}\) of all pre-bipartite digraphs containing no cycles is a universal Horn class. Let \({\mathcal{U}}\) be a universal Horn class contained in \({\mathcal{B}}\). We determine when it is possible to axiomatise, by first-order sentences, the class \({\mathcal{R}_{\rm CT}(\mathcal{U}_{\rm fin})}\) of compact topological digraphs that are topologically residually in the class of finite members of \({\mathcal{U}}\). We show that if \({\mathcal{R}_{\rm CT}(\mathcal{U}_{\rm fin})}\) is axiomatisable by first-order sentences, then it is axiomatisable by universal Horn sentences.