Ideal lattices of lattices

This paper shows that any compactly generated lattice is a subdirect product of subdirectly irreducible lattices which are complete and upper continuous. An example of a compactly generated lattice which cannot be subdirectly decomposed into subdirectly irreducible compactly generated lattices is given. In the case of an ideal lattice of a lattice L, the decomposition into subdirectly irreducible complete lattices is tied, via a special completion process, to the finitely subdirectly irreducible homomorphic of images L. It is also shown that any finite lattice satisfying the Whitman condition is a retract of the ideal lattice of the dual ideal lattice of a free lattice.