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.
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.
@article{CMJ_2001_51_2_a12,
author = {Steinberg, Stuart A.},
title = {Finitely valued $f$-modules, an addendum},
journal = {Czechoslovak Mathematical Journal},
pages = {387--394},
year = {2001},
volume = {51},
number = {2},
mrnumber = {1844318},
zbl = {0979.06010},
language = {en},
url = {http://geodesic.mathdoc.fr/item/CMJ_2001_51_2_a12/}
}
TY - JOUR
AU - Steinberg, Stuart A.
TI - Finitely valued $f$-modules, an addendum
JO - Czechoslovak Mathematical Journal
PY - 2001
SP - 387
EP - 394
VL - 51
IS - 2
UR - http://geodesic.mathdoc.fr/item/CMJ_2001_51_2_a12/
LA - en
ID - CMJ_2001_51_2_a12
ER -
%0 Journal Article
%A Steinberg, Stuart A.
%T Finitely valued $f$-modules, an addendum
%J Czechoslovak Mathematical Journal
%D 2001
%P 387-394
%V 51
%N 2
%U http://geodesic.mathdoc.fr/item/CMJ_2001_51_2_a12/
%G en
%F CMJ_2001_51_2_a12
Steinberg, Stuart A. Finitely valued $f$-modules, an addendum. Czechoslovak Mathematical Journal, Tome 51 (2001) no. 2, pp. 387-394. http://geodesic.mathdoc.fr/item/CMJ_2001_51_2_a12/
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