Definability in substructure orderings, IV: Finite lattices

Let $${\mathcal{L}}$$ be the ordered set of isomorphism types of finite lattices, where the ordering is by embeddability. We study first-order definability in this ordered set. Our main result is that for every finite lattice L, the set {ℓ, ℓopp} is definable, where ℓ and ℓopp are the isomorphism types of L and its opposite (L turned upside down). We shall show that the only non-identity automorphism of $${\mathcal{L}}$$ is the map $${\ell \mapsto \ell^{\rm opp}}$$.