Laterally closed lattice homomorphisms

[摘要] 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. (c) 2006 Elsevier Inc. All rights reserved.

[发布日期] 2006-12-15

[关键词] laterally closed lattice homomorphism;d-algebra;f-algebra