Laterally closed lattice homomorphisms

Abstract

Let A and B be two Archimedean vector lattices and let T : A → B be a lattice homomorphism. We call that T is laterally closed if T ( D ) is a maximal orthogonal system in the band generated by T ( A ) in B, for each maximal orthogonal system D of A. In this paper we prove that any laterally closed lattice homomorphism T of an Archimedean vector lattice A with universal completion A u into a universally complete vector lattice B can be extended to a lattice homomorphism of A u into B, which is an improvement of a result of M. Duhoux and M. Meyer [M. Duhoux and M. Meyer, Extended orthomorphisms and lateral completion of Archimedean Riesz spaces, Ann. Soc. Sci. Bruxelles 98 (1984) 3–18], who established it for the order continuous lattice homomorphism case. Moreover, if in addition A u and B are with point separating order duals ( A u ) ′ and B ′ respectively, then the laterally closedness property becomes a necessary and sufficient condition for any lattice homomorphism T : A → B to have a similar extension to the whole A u . As an application, we give a new representation theorem for laterally closed d-algebras from which we infer the existence of d-algebra multiplications on the universal completions of d-algebras.