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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.
06F25 Ordered rings, algebras, modules
46A40 Ordered topological linear spaces, vector lattices
47B65 Positive linear operators and order-bounded operators
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