Definability in substructure orderings, III: Finite distributive lattices

Abstract

Let \({{\mathcal D}}\) be the ordered set of isomorphism types of finite distributive lattices, where the ordering is by embeddability. We study first-order definability in this ordered set. We prove among other things that for every finite distributive lattice D, the set {d, dopp} is definable, where d and dopp are the isomorphism types of D and its opposite (D turned upside down). We prove that the only non-identity automorphism of \({{\mathcal D}}\) is the opposite map. Then we apply these results to investigate definability in the closely related lattice of universal classes of distributive lattices. We prove that this lattice has only one non-identity automorphism, the opposite 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, Kopp} is a definable subset of the lattice.