SUBLATTICES OF LATTICES OF ORDER-CONVEX SETS, III: THE CASE OF TOTALLY ORDERED SETS

摘要

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 lattice of the form Π_(i∈I)Co(T_i), where is a family 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 finite, finitely based varieties of lattices, and we find finite equational bases of these varieties. (2) The variety SUB(LO) is the quasivariety join of all the varieties SUB(n), for 1≤n<ω, and it has only countably many subvarieties. We classify these varieties, together with all the finite subdirectly irreducible members of SUB(LO). (3) Every finite subdirectly irreducible member of SUB(LO) is projective within SUB(LO), and every subquasivariety of SUB(LO) is a variety.