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A partial lattice P is ideal-projective, with respect to a class C oflattices, if for every K $\in$ C and every homomorphism $\phi$ of partiallattices from P to the ideal lattice of K, there are arbitrarily large choicefunctions f : P $\rightarrow$ K for $\phi$ that are also homomorphisms ofpartial lattices. This extends the traditional concept of (sharp)transferability of a lattice with respect to C. We prove the following: (1) Afinite lattice P, belonging to a variety V, is sharply transferable withrespect to V iff it is projective with respect to V and weakly distributivelattice homomorphisms, iff it is ideal-projective with respect to V. (2) Everyfinite distributive lattice is sharply transferable with respect to the class Rmod of all relatively complemented modular lattices. (3) The gluing D 4 of twosquares, the top of one being identified with the bottom of the other one, issharply transferable with respect to a variety V iff V is contained in thevariety 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, butnot sharply transferable, with respect to the variety M of all modularlattices. This solves a 1978 problem of G. Gr\"atzer. (6) We construct amodular lattice whose canonical embedding into its ideal lattice is not pure.This solves a 1974 problem of E. Nelson.

Relative projectivity and transferability for partial lattices