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 [Formula: see text] (resp., SUB(n)) the class of all lattices that can be embedded into a lattice of the form [Formula: see text] where <Ti|i∈I> is a family of chains (resp., chains with at most n elements). We prove the following results:(1) Both classes [Formula: see text] 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 [Formula: see text] 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 [Formula: see text].(3) Every finite subdirectly irreducible member of [Formula: see text] is projective within [Formula: see text], and every subquasivariety of [Formula: see text] is a variety.
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