Infinite independent sets in distributive lattices

We show that a poset P contains a subset isomorphic toif and only if the poset J(P) consisting of ideals of P contains a subset isomorphic to 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 or as sublattices, where Γ and Δ are two special meet-semilattices already considered by J. D. Lawson, M. Mislove and H. A. Priestley.Dedicated to the memory of Ivan RivalReceived April 22, 2003; accepted in final form July 11, 2004.This revised version was published online in August 2005 with a corrected cover date.