In the paper we prove that every orthosymmetric lattice bilinear map on the cartesian product of a vector lattice with itself can be extended to an orthosymmetric lattice bilinear map on the cartesian product of the Dedekind completion with itself. The main tool used in our proof is the technique associated with extension to a vector subspace generated by adjoining one element. As an application, we prove that if (A, ∗) is a commutative d-algebra and A δ its Dedekind completion, then, A δ can be equipped with a d-algebra multiplication that extends the multiplication of A. Moreover, we indicate an error made in the main result of the paper: M. A. Toumi, Extensions of orthosymmetric lattice bimorphisms, Proc. Amer. Math. Soc. 134 (2006), 1615–1621.
The main topic of the first section of this paper is the following theorem: let A be an Archimedean f-algebra with unit element e, and T : A → A a Riesz homomorphism such that T 2 (f) = T(fT(e)) for all f ∈ A. Then every Riesz homomorphism extension Te of T from the Dedekind completion A δ of A into itself satisfies Te2 (f) = Te(fT(e)) for all f ∈ A δ . In the second section this result is applied in several directions. As a first application it is applied to show a result about extensions of positive projections to the Dedekind completion. A second application of the above result is a new approach to the Dedekind completion of commutative d-algebras.