Arens regularity of lattice-ordered rings
Let \(R\) be a unital lattice-ordered ring, and let \(R^{\star}\) = Hom\(_{b}(R, \mathbb{R})\) be the set of all bounded additive group homomorphisms from \(R\) to the field \(\mathbb{R}\) of real numbers. Define \((f + g)(u) := f(u) + g(u)\), \(\forall f, g \in R^{\star}\), and \((R^{\star})^{+} := \{f \in R^{\star} \mid f(R^{+}) \subseteq \mathbb{R}^{+}\}\). Then \(R^{\star}\) becomes a complete abelian lattice-ordered group, which is called the order conjugate of \(R\). The order conjugate \(R^{\star\star} = (R^{\star})^{\star}\) of \(R^\star\) is called the order biconjugate of \(R\). From constructions given by \textit{R. Arens} [Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)], two multiplications on \(R^{\star\star}\) were introduced as follows. \[ \forall \varphi, \psi \in R^{\star\star}\;\forall f \in R^{\star}\;(\varphi \ltimes \psi)(f) = \varphi(\psi f),\tag{1} \] where \(\psi f \in R^{\star}\) is defined by \(\forall u \in R\) \((\psi f)(u) = \psi(fu)\), and \(fu \in R^{\star}\) is defined by \(\forall v \in R\) \((fu)(v) = f(uv)\). The operation \(\ltimes\) is called the first Arens multiplication on \(R^{\star\star}\), and \(R^{\star\star}\) becomes a lattice-ordered ring with respect to the multiplication \(\ltimes\). \[ \forall \varphi, \psi \in R^{\star\star}\;\forall f \in R^{\star} \;(\varphi \rtimes \psi)(f) = \psi(f\varphi),\tag{2} \] where \(f\varphi \in R^{\star}\) is defined by \(\forall u \in R\) \((f\varphi)(u) = \varphi(uf)\), and \(uf \in R^{\star}\) is defined by \(\forall v \in R\) \((uf)(v) = f(vu)\). The operation \(\rtimes\) on \(R^{\star\star}\) is called the second Arens multiplication on \(R^{\star\star}\), and \(R^{\star\star}\) becomes a lattice-ordered ring with respect to the multiplication \(\rtimes\). A unital lattice-ordered ring \(R\) is said to be Arens regular if \(\ltimes = \rtimes\) on \(R^{\star\star}\). The article proves that for any unital almost \(f\)-ring \(R\), \(R\) is Arens regular and \(R^{\star\star}\) is an \(f\)-ring. An example is provided to show that a general unital lattice-ordered ring may not be Arens regular.
scientific article; zbMATH DE number 5799079
scientific article