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On extensions of orthosymmetric lattice bimorphisms. (English) Zbl 1289.06029
Let $$A$$ be an Archimedean Riesz space (i.e., an Archimedean vector lattice) with $$A^{\delta }$$ as Dedekind completion. A bilinear map $$a\: A\times A\rightarrow A$$ is said to be orthosymmetric if $$a\left ( x,y\right )=0$$ whenever $$x\wedge y=0$$. Moreover, $$a$$ is called a lattice bilinear map if it is a lattice homomorphism in each variable. In this paper, the author proves that every orthosymmetric lattice bilinear map $$A\times A\rightarrow A$$ has an extension $$a^{\delta }\: A^{\delta }\times A^{\delta }\rightarrow A^{\delta }$$ which is again an orthosymmetric lattice bilinear map. As an application, he concludes that if $$A$$ is equipped with a commutative $$d$$-algebra multiplication $$\ast$$, then $$A^{\delta }$$ is endowed with a $$d$$-algebra multiplication that extends $$\ast$$. By the way, the author points out that the main result of his previous note [Proc. Am. Math. Soc. 134, No. 6, 1615–1621 (2006; Zbl 1100.06013)] is wrong and he gives a counterexample for that.
MSC:
 06F25 Ordered rings, algebras, modules 46A40 Ordered topological linear spaces, vector lattices 47B65 Positive linear operators and order-bounded operators
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