Extensions of orthosymmetric lattice bimorphisms

Let $E$ be an Archimedean vector lattice, let $E^{\mathfrak{d}}$ be its Dedekind completion and let $B$ be a Dedekind complete vector lattice. If $\Psi _{0}:E\times E\rightarrow B$ is an orthosymmetric lattice bimorphism, then there exists a lattice bimorphism $\Psi:E^{\mathfrak{d}}\times E^{\mathfrak{d}} \rightarrow B$ that not just extends $\Psi_{0}$ but also has to be orthosymmetric. As an application, we prove the following: Let $A$ be an Archimedean $d$-algebra. Then the multiplication in $A$ can be extended to a multiplication in $A^{\mathfrak{d}}$, the Dedekind completion of $A$, in such a fashion that $A^{\mathfrak{d}}$ is again a $d$-algebra with respect to this extended multiplication. This gives a positive answer to the problem posed by C. B. Huijsmans in 1990.