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

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 < \omega, 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.