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

Semenova, Marina V.; Wehrung, Friedrich

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

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Description

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 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.