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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)].
##### MSC:
 06F15 Ordered groups 06F25 Ordered rings, algebras, modules
##### Keywords:
lattice-ordered module; value set; lattice-ordered group
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##### References:
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