Extensions of orthosymmetric lattice bimorphisms

Toumi, Mohamed

doi:10.1090/S0002-9939-05-08142-6

Extensions of orthosymmetric lattice bimorphisms
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Proc. Amer. Math. Soc. 134 (2006), 1615-1621 Request permission

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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