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Extensions of orthosymmetric lattice bimorphisms

Author: Mohamed Ali Toumi
Journal: Proc. Amer. Math. Soc. 134 (2006), 1615-1621
MSC (2000): Primary 06F25, 47B65
Published electronically: December 5, 2005
MathSciNet review: 2204271
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Abstract: Let $ E$ be an Archimedean vector lattice, let $ E^{\mathfrak{d}}$ be its Dedekind completion and let $ B$ be a Dedekind complete vector lattice. If $ \Psi _{0}:E\times E\rightarrow B$ is an orthosymmetric lattice bimorphism, then there exists a lattice bimorphism $ \Psi:E^{\mathfrak{d}}\times E^{\mathfrak{d}} \rightarrow B$ that not just extends $ \Psi_{0}$ but also has to be orthosymmetric. As an application, we prove the following: Let $ A$ be an Archimedean $ d$-algebra. Then the multiplication in $ A$ can be extended to a multiplication in $ A^{\mathfrak{d}}$, the Dedekind completion of $ A$, in such a fashion that $ A^{\mathfrak{d}}$ is again a $ d$-algebra with respect to this extended multiplication. This gives a positive answer to the problem posed by C. B. Huijsmans in 1990.

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Mohamed Ali Toumi
Affiliation: Département des Mathématiques, Faculté des Sciences de Bizerte, 7021 Zarzouna, Bizerte, Tunisia

Keywords: $d$-algebra, $f$-algebra, lattice homomorphism, lattice bimorphism
Received by editor(s): February 10, 2004
Received by editor(s) in revised form: January 13, 2005
Published electronically: December 5, 2005
Additional Notes: The author thanks Professor S. J. Bernau for providing the bibliographic information of [2]
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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