Extension of Vector Lattice Homomorphisms

Abstract

Suppose that B is a complete Boolean algebra, D a distributive lattice and φ a lattice homomorphism from D0, a sublattice of D, into B; then φ can be extended to a lattice homomorphism of D into B. This generalizes Sikorski's extension theorem for Boolean algebras. It also leads to a new proof that if N is a complete vector lattice, L a vector lattice, M a subvector lattice of L, and f a vector lattice homomorphism of M into N, then f can be extended to a vector lattice homomorphism, g, say, of L into N. The proof of this is constructive after applying the lattice homomorphism extension theorem, and leads to the previously unknown result that g is uniquely determined by the collection of polar subspaces, g(x)⊥⊥ (x ∈ L). The paper concludes with a new approach to the known result that the vector lattice homomorphic extensions of f are precisely the extreme points of the set of positive extensions of f.