Relative projectivity and transferability for partial lattices

Abstract: A partial lattice P is ideal-projective, with respect to a class C of lattices, if for every K $\in$ C and every homomorphism $\phi$ of partial lattices from P to the ideal lattice of K, there are arbitrarily large choice functions f : P $\rightarrow$ K for $\phi$ 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 R mod of all relatively complemented modular lattices. (3) The gluing D 4 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 M$\omega$ generated by all lattices of length 2. (4) D 4 is projective, but not ideal-projective, with respect to R mod. (5) D 4 is transferable, but not sharply transferable, with respect to the variety M of all modular lattices. This solves a 1978 problem of G. Gr\"atzer. (6) We construct a modular lattice whose canonical embedding into its ideal lattice is not pure. This solves a 1974 problem of E. Nelson.