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DEFINABILITY IN SUBSTRUCTURE ORDERINGS, II: FINITE ORDERED SETS

By Jaroslav Je, Zek and Ralph Mckenzie

Abstract

Abstract. 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 member K of either of these two definable 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 finite posets that is definable in the full second-order language over the domain of finite posets is, after factoring by isomorphism, firstorder definable up to duality in the ordered set P. 1

Year: 2013
OAI identifier: oai:CiteSeerX.psu:10.1.1.298.974
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