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Finitely valued \(f\)-modules, an addendum. (English) Zbl 0979.06010
Summary: In an \(\ell\)-group \(M\) with an appropriate operator set \(\Omega\) it is shown that the \(\Omega\)-value set \(\Gamma_{\Omega}(M)\) can be embedded in the value set \(\Gamma(M)\). This embedding is an isomorphism if and only if each convex \(\ell\)-subgroup is an \(\Omega\)-subgroup. If \(\Gamma(M)\) has a.c.c. and \(M\) is either representable or finitely valued, then the two value sets are identical. More generally, these results hold for two related operator sets \(\Omega_1\) and \(\Omega_2\) and the corresponding \(\Omega\)-value sets \(\Gamma_{\Omega_1}(M)\) and \(\Gamma_{\Omega_2}(M)\). If \(R\) is a unital \(\ell \)-ring, then each unital \(\ell\)-module over \(R\) is an \(f\)-module and has \(\Gamma(M) = \Gamma_R(M)\) exactly when \(R\) is an \(f\)-ring in which \(1\) is a strong order unit.
The title of this article refers to the author’s paper in Pac. J. Math. 40, 723-737 (1972; Zbl 0218.16008)].
06F15 Ordered groups
06F25 Ordered rings, algebras, modules
Full Text: DOI EuDML
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