# Article

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Keywords:
lattice-ordered module; value set
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.
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