2005/01/21 by Marina V. Semenova, Friedrich Wehrung · 1 citation
Mathematics · #math.GM #msc:06B05 #msc:06B15 #msc:06B23 #msc:08C15. #msc:05B25 #msc:05C05
published as Journal of Algebra 277, no. 2 (2004) 825--860
arxiv created 2005/01/21 · arxiv updated 2009/12/01
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 2n2-5n+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.