Finitely valued $f$-modules, an addendum

Steinberg, Stuart A.

Czechoslovak Mathematical Journal, Tome 51 (2001) no. 2, pp. 387-394.

Résumé

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.

Détail

Classification : 06F15, 06F25
Keywords: lattice-ordered module; value set