Laterally closed lattice homomorphisms

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 Au into a universally complete vector lattice B can be extended to a lattice homomorphism of Au 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 Au and B are with point separating order duals (Au) 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 Au. 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. © 2006 Elsevier Inc. All rights reserved