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Sublattices of lattices of order-convex sets, I. The main representation theorem

Marina V. Semenova·Friedrich Wehrung
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Abstract

For a partially ordered set P, we denote by Co(P) the lattice of order-convex subsets of P. We find three new lattice identities, (S), (U), and (B), such that the following result holds. Theorem. Let L be a lattice. Then L embeds into some lattice of the form Co(P) iff L satisfies (S), (U), and (B). Furthermore, if L has an embedding into some Co(P), then it has such an embedding that preserves the existing bounds. If L is finite, then one can take P finite, of cardinality at most $2n^2-5n+4$, where n is the number of join-irreducible elements of L. On the other hand, the partially ordered set P can be chosen in such a way that there are no infinite bounded chains in P and the undirected graph of the predecessor relation of P is a tree.

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Marina V. Semenova, Friedrich Wehrung. 2005-01-21. Sublattices of lattices of order-convex sets, I. The main representation theorem. https://arxiv.org/abs/math/0501341

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