The poset of morphism-extension classes of countable graphs

Abstract Let XY L , T consist of all countable -structures M that satisfy the axioms and in which homomorphisms type X (these could be plain homomorphisms, monomorphisms, or isomorphisms) between finite substructures are restrictions an endomorphism Y (for example, automorphism a surjective endomorphism). Lockett Truss introduced 18 such classes for relational structures. For given pair however, two more morphism-extension properties may define same class In this paper, we establish equalities inequalities graphs.