Extensions of orthosymmetric lattice bimorphisms revisited

Abstract:

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.