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Extensions of orthosymmetric lattice bimorphisms revisited

Authors:	Karim Boulabiar

Institution:	Institut Préparatoire aux Etudes Scientifiques et Techniques, Université 7 novembre à Carthage, BP51, 2070-La Marsa, Tunisia

Abstract:	This note furnishes an example illustrating the following two facts. On the one hand, there exist Archimedean Riesz spaces and with 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 . On the other hand, there is an associative -multiplication $\ast$ in the same Archimedean Riesz space which extends to a -multiplication $\ast^{\delta}$ in $E^{\delta}$ which is not associative. The existence of such an example provides counterexamples to assertions in Toumi, 2005.

Keywords:	Extension Dedekind-completion $d$-multiplication lattice bimorphism orthosymmetric Riesz space

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