Relative Projectivity and Transferability for Partial Lattices

Abstract

A partial lattice P is ideal-projective, with respect to a class $\mathcal {C}$ of lattices, if for every $K\in \mathcal {C}$ 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 $\mathcal {C}$. We prove the following: (1) A finite lattice P, belonging to a variety $\mathcal {V}$, is sharply transferable with respect to $\mathcal {V}$ iff it is projective with respect to $\mathcal {V}$ and weakly distributive lattice homomorphisms, iff it is ideal-projective with respect to $\mathcal {V}$, (2) Every finite distributive lattice is sharply transferable with respect to the class $\mathcal {R}_{\text {mod}}$ 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 $\mathcal {V}$ iff $\mathcal {V}$ is contained in the variety $\mathcal {M}_{\omega }$ generated by all lattices of length 2, (4) D ₄ is projective, but not ideal-projective, with respect to $\mathcal {R}_{\text {mod}}$ , (5) D ₄ is transferable, but not sharply transferable, with respect to the variety $\mathcal {M}$ 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.