Relative Projectivity and Transferability for Partial Lattices

A partial lattice P is ideal-projective, with respect to a class C of lattices, if for every K ∈ C and every homomorphism φ of partial lattices from P to the ideal lattice of K, there are arbitrarily large choice functions f : P → K for φ that are also homomorphisms of partial lattices. This extends the traditional concept of (sharp) transferability of a lattice with respect to C. We prove the following: (1) A finite lattice P , belonging to a variety V, is sharply transferable with respect to V iff it is projective with respect to V and weakly distributive lattice homomorphisms, iff it is ideal-projective with respect to V. (2) Every finite distributive lattice is sharply transferable with respect to the class Rmod of all relatively complemented modular lattices. (3) The gluing D4 of two squares, the top of one being identified with the bottom of the other one, is sharply transferable with respect to a variety V iff V is contained in the variety Mω generated by all lattices of length 2. (4) D4 is projective, but not ideal-projective, with respect to Rmod . (5) D4 is transferable, but not sharply transferable, with respect to the variety M of all modular lattices. This solves a 1978 problem of G. Grätzer. (6) We construct a modular lattice whose canonical embedding into its ideal lattice is not pure. This solves a 1974 problem of E. Nelson.