Cores and compactness of infinite directed graphs
We define the property of homomorphic compactness for digraphs. We prove that if a digraph \(H\) is homomorphically compact then \(H\) has a core, although the converse does not hold. We also examine a weakened compactness condition and show that when this condition is assumed, compactness is equivalent to containing a core. We use this result to prove that if a digraph \(H\) of size \(\kappa\) is not compact, then there is a digraph \(G\) of size at most \(\kappa^+\) such that \(H\) is not compact with respect to \(G\). We then give examples of some sufficient conditions for compactness.
scientific article; zbMATH DE number 954827