Some Homological Pathology in Vector Lattices

Abstract

The purpose of this paper is to point out a number of curious phenomena in the category of (real) vector lattices and linear lattice homomorphisms. Birkhoff (3, p. 221, Ex. 2 and Problem 96) called attention to the question of constructing models of the free objects with more than one generator in this category, a problem recently solved by E. C. Weinberg (9). In §6 we construct a more manageable class of (non-free) projective vector lattices. Here, however, there is a countability restriction which suggests strong connections with free and projective Boolean algebras (in the category of Boolean algebras and their homomorphisms, such algebras must satisfy the countable chain condition (6)).