Sublattices of lattices of order-convex sets, II. Posets of finite length

2005/01/21 by Marina V. Semenova, Friedrich Wehrung
Mathematics · #math.GM #msc:06B05 #msc:06B15 #msc:06B23 #msc:08C15. #msc:05B25 #msc:05C05

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published as International Journal of Algebra and Computation 13, no. 5 (2003) 543-564

arxiv created 2005/01/21 · arxiv updated 2009/12/01

Abstract

For a positive integer n, we denote by SUB (resp., SUBn) the class of all lattices that can be embedded into the lattice Co(P) of all order-convex subsets of a partially ordered set P (resp., P of length at most n). We prove the following results: (1) SUBn is a finitely based variety, for any n ≥ 1. (2) SUB2 is locally finite. (3) A finite atomistic lattice L without D-cycles belongs to SUB iff it belongs to SUB2; this result does not extend to the nonatomistic case. (4) SUBn is not locally finite for n ≥ 3.

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