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

Marina V. Semenova (1) , Friedrich Wehrung (2)

1 Institute of Mathematics
2 LMNO - Laboratoire de Mathématiques Nicolas Oresme

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.

Mots clés: join-irreducible; Lattice; embedding; poset; order-convex; join-semidistributivity; 2- distributivity; join-irreducible.

Domaines: Mathématiques générales [math.GM]

Marina V. Semenova, Friedrich Wehrung. Sublattices of lattices of order-convex sets, I. The main representation theorem. Journal of Algebra, 2004, 277 (2), pp.825--860. ⟨10.1016/j.jalgebra.2004.01.023⟩. ⟨hal-00003980⟩