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

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.