Abstract
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, ℓ opp} is definable, where ℓ and ℓ opp 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 \({\ell \mapsto \ell^{\rm opp}}\).
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Presented by M. Valeriote.
While working on this paper, the authors were supported by US NSF grant DMS-0604065. The first author (Ježek) was also supported by the Grant Agency of the Czech Republic, grant #201/05/0002 and by the institutional grant MSM0021620839 financed by MSMT.
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Ježek, J., McKenzie, R. Definability in substructure orderings, IV: Finite lattices. Algebra Univers. 61, 301 (2009). https://doi.org/10.1007/s00012-009-0019-x
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DOI: https://doi.org/10.1007/s00012-009-0019-x