Extensions of orthosymmetric lattice bimorphisms revisited

This note furnishes an example illustrating the following two facts. On the one hand, there exist Archimedean Riesz spaces $E$ and $F$ with $F$ Dedekind-complete and an orthosymmetric lattice bimorphism $\Psi:E\times E\rightarrow F$ with lattice bimorphism extension $\Psi^{\delta}:E^{\delta }\times E^{\delta}\rightarrow F$ which is not orthosymmetric, where $E^{\delta}$ denotes the Dedekind-completion of $E$. On the other hand, there is an associative $d$-multiplication $\ast$ in the same Archimedean Riesz space $E$ which extends to a $d$-multiplication $\ast^{\delta}$ in $E^{\delta}$ which is not associative. The existence of such an example provides counterexamples to assertions in Toumi, 2005.