Sublattices of lattices of order-convex sets, III. The case of totally ordered sets

2005/01/21 by Marina V. Semenova, Friedrich Wehrung
Mathematics · #math.GM #msc:06B05 #msc:06B20 #msc:06B15 #msc:06A05 #msc:08C15. #msc:05B25

paper · pdf

published as International Journal of Algebra and Computation 14, no. 3 (2004) 357-387

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

Abstract

For a partially ordered set P, let Co(P) denote the lattice of all order-convex subsets of P. For a positive integer n, we denote by SUB(LO) (resp., SUB(n)) the class of all lattices that can be embedded into a product of lattices of convex subsets of chains (resp., chains with at most n elements). We prove the following results: (1) Both classes SUB(LO) and SUB(n), for any positive integer n, are locally &#64257;nite, &#64257;nitely based varieties of lattices, and we &#64257;nd &#64257;nite equational bases of these varieties. (2) The variety SUB(LO) is the quasivariety join of all the varieties SUB(n), for 1 &#8804; n < ω, and it has only countably many subvarieties. We classify these varieties, together with all the &#64257;nite subdirectly irreducible members of SUB(LO). (3) Every &#64257;nite subdirectly irreducible member of SUB(LO) is projective within SUB(LO), and every subquasivariety of SUB(LO) is a variety.

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