In this paper, we present some results concerning the automatic order boundedness of band preserving operators on Dedekind σ -complete vector lattices.

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.

Let A, B be two archimedean -algebras and let U,V be two positive linear maps from A to B. We call that the couple (U,V ) is separating with respect to A and B if |a| |b| = 0 in A implies |U (a)||V (b)| = 0 in B. In this paper, we prove that if A is an f -algebra with unit elment e, if B is an -algebra and if (U,V ) is a separating couple with respect to A and B then (U ∼∼ ,V ∼∼ ) , where U ∼∼ (resp V ∼∼ ) is the bi-adjoint of U (resp of V ), is again a separating couple with respect to the order continuous order biduals (A ) n and (B ) n of A and B respectively furnished with their Arens products respectively. Moreover, in the case where B separates the points of B, we give a characterization of any separating couple with respect to A and B.

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The paper presents simple proofs of the Cauchy-Schwartz inequality and the negative discriminant property in archimedean almost f -algebras [5] , based on a sequence approximation.