Let 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} is definable, where l and l are the isomorphism types of L and its opposite (L turned upside down). We shall show that the only non-identity automorphism of L is the map...

LACES DEFINITION. Let P be a finite set of properties. A mapping l that Ž . assigns to any subset S ; P another subset l S is called a lace map if, for all S, G, S , S ; P: 1 2 Ž . Ž . i l S ; S; Ž . Ž . Ž . Ž . ii l S ; G ; S « l G s l S ; Ž . Ž . Ž . Ž . Ž . iii l S s l S « l S j S s l S . 1 2 1 2 1 Ž . Ž . Ž . A set L for which l L s L is called a lace. By applying ii to G s l S , Ž Ž .. Ž ....

David Brydges and Thomas Spencer's Lace Expansion is abstracted, and it is shown how it sometimes gives rise to sieves. LACES Definition: Let P be a finite set of properties. A mapping l that assigns to any subset S ⊂ P another subset l(S), is called a lace-map, A set L for which l(L) = L is called a lace. By applying (ii) to G = l(S), it is seen that l(l(S)) = l(S), for any set of properties S...

Every lattice is isomorphic to a lattice whose elements are sets of sets and whose operations are intersection and the operation ∨∗ defined by A ∨∗ B = A ∪ B ∪ {Z : (∃X ∈ A)(∃Y ∈ B)X ∩ Y ⊆ Z}. This representation spells out precisely Birkhoff’s and Frink’s representation of arbitrary lattices, which is related to Stone’s set-theoretic representation of distributive lattices. (AMS Subject Classi...