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Extensions of orthosymmetric lattice bimorphisms

Authors:	Mohamed Ali Toumi

Institution:	Département des Mathématiques, Faculté des Sciences de Bizerte, 7021 Zarzouna, Bizerte, Tunisia

Abstract:	Let be an Archimedean vector lattice, let $E^{\mathfrak{d}}$ be its Dedekind completion and let 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 be an Archimedean -algebra. Then the multiplication in can be extended to a multiplication in $A^{\mathfrak{d}}$ , the Dedekind completion of , in such a fashion that $A^{\mathfrak{d}}$ is again a -algebra with respect to this extended multiplication. This gives a positive answer to the problem posed by C. B. Huijsmans in 1990.

Keywords:	$d$-algebra $f$-algebra lattice homomorphism lattice bimorphism

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