Definability in substructure orderings. II: Finite ordered sets
In this paper the authors study first-order definability in the ordered set \( \mathcal{P}\) of isomorphism types of finite ordered sets (posets), where the ordering is by embeddability. They prove, among other things, that for every finite poset \(P\), the set \(\{p,p^{\partial }\}\) is definable, where \(p\) and \( p^{\partial }\) are the isomorphism types of \(P\) and its dual poset. Also, they prove that the only non-identity automorphism of \(\mathcal{P}\) is the duality map. Then they apply these results to investigate definability in the closely related lattice of universal classes of posets. Also, they prove that this lattice has only one non-identity automorphisms, the duality map; that the set of finitely generated universal classes as well as the set of finitely axiomatizable universal classes are definable subsets of the lattice; and that, for each member \(K\) of either of these two definable subsets, a definable subset \(\{K,K^{\partial }\}\) is a definable subset of the lattice. For Part I see Algebra Univers. 61, No.~1, 59--75 (2009; Zbl 1181.08004).
isomorphism types of finite ordered sets
scientific article; zbMATH DE number 5729753