Definability in Substructure Orderings, II: Finite Ordered Sets

Let P be the ordered set of isomorphism types of finite ordered sets (posets), where the ordering is by embeddability. We study first-order definability in this ordered set. We prove among other things that for every finite poset P, the set {p, p∂ } is definable, where p and p∂ are the isomorphism types of P and its dual poset. We prove that the only non-identity automorphism of P is the duality map. Then we apply these results to investigate definability in the closely related lattice of universal classes of posets. We prove that this lattice has only one non-identity automorphism, the duality map; that the set of finitely generated and also the set of finitely axiomatizable universal classes are definable subsets of the lattice; and that for each element K of the two subsets, {K, K∂ } is a definable subset of the lattice. Next, making fuller use of the techniques developed to establish these results, we go on to show that every isomorphism-invariant relation between �...

• Type: Journal
• Year: 2010
• Where: ORDER
• Authors: Jaroslav Jezek, Ralph McKenzie
• Keywords: Duality Map, Isomorphism Types, Non-identity Automorphism, ORDER 2010