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Extensions of orthosymmetric lattice bimorphisms revisited

Author: Karim Boulabiar
Journal: Proc. Amer. Math. Soc. 135 (2007), 2007-2009
MSC (2000): Primary 06F25, 47B65
Published electronically: February 6, 2007
MathSciNet review: 2299473
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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.

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

Keywords: Extension, Dedekind-completion, $d$-multiplication, lattice bimorphism, orthosymmetric, Riesz space.
Received by editor(s): March 8, 2006
Received by editor(s) in revised form: April 19, 2006
Published electronically: February 6, 2007
Additional Notes: The author would like to thank the referee for his helpful suggestions and comments which considerably improved preliminary versions of this work.
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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