Infinite independent sets in distributive lattices

Abstract

We show that a poset P contains a subset isomorphic to \([\kappa ]^{ < \omega }\)if and only if the poset J(P) consisting of ideals of P contains a subset isomorphic to \(\mathcal{P}(\kappa ),\) the power set of κ. If P is a join-semilattice this amounts to the fact that P contains an independent set of size κ. We show that if κ := ω and P is a distributive lattice, then this amounts to the fact that P contains either \(I_{ < \omega } (\Gamma )\) or \(I_{ < \omega } (\Delta )\) as sublattices, where Γ and Δ are two special meet-semilattices already considered by J. D. Lawson, M. Mislove and H. A. Priestley.