Abstract
This chapter presents a report on sublattices of a free lattice. It elaborates the characterization problem for finite sublattices of a free lattice. It is conjectured that these are precisely the finite lattices that satisfy three first order conditions, the Whitman condition. Every finite sublattice of a free lattice is an S-lattice, for all three conditions hold in free lattices, and is the key condition in Whitmans famous characterization of free lattices, and is easy consequences of Whitmans results. It is an open question whether, conversely, every S-lattice is embeddable in a free lattice. Several characterizations of finite sublattices of free lattices have been found and some of these even characterize finitely generated sublattices of free lattices, but none of them is in terms of a finite set of first-order formulas. A lattice L is transferable if for every embedding f of L into the ideal lattice of a lattice K, there exists and embedding g of L into K.
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