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 E and F F with F F Dedekind-complete and an orthosymmetric lattice bimorphism Ψ : E × E → F \Psi :E\times E\rightarrow F with lattice bimorphism extension Ψ δ : E δ × E δ → F \Psi ^{\delta }:E^{\delta }\times E^{\delta }\rightarrow F which is not orthosymmetric, where E δ E^{\delta } denotes the Dedekind-completion of E E . On the other hand, there is an associative d d -multiplication ∗ \ast in the same Archimedean Riesz space E E which extends to a d d -multiplication ∗ δ \ast ^{\delta } in E δ E^{\delta } which is not associative. The existence of such an example provides counterexamples to assertions in Toumi, 2005.