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Extensions of orthosymmetric lattice bimorphisms. (English) Zbl 1100.06013
For an Archimedian vector lattice $$E$$ and a Dedekind complete vector lattice $$B$$, it is shown that if a lattice bimorphism from $$E\times E$$ to $$B$$ satisfies the AF property (the image of each pair of disjoint elements is zero), then every lattice extension to the cartesian product of the Dedekind completion of $$E$$ with itself also satisfies the AF property. As a consequence, the author demonstrates that the multiplication in an Archimedean d-algebra can be extended to a d-algebra multiplication in the Dedekind completion. This result was established by E. Chil [Positivity 8, 257–267 (2004; Zbl 1073.46002)] but is proved here in a brief and direct manner.

##### MSC:
 06F20 Ordered abelian groups, Riesz groups, ordered linear spaces 46A40 Ordered topological linear spaces, vector lattices 06F25 Ordered rings, algebras, modules 47B65 Positive linear operators and order-bounded operators
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