Finitely valued $f$-modules, an addendum

Steinberg, Stuart A.. "Finitely valued $f$-modules, an addendum." Czechoslovak Mathematical Journal 51.2 (2001): 387-394. <http://eudml.org/doc/30642>.

@article{Steinberg2001,
abstract = {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.},
author = {Steinberg, Stuart A.},
journal = {Czechoslovak Mathematical Journal},
keywords = {lattice-ordered module; value set; lattice-ordered module; value set; lattice-ordered group},
language = {eng},
number = {2},
pages = {387-394},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Finitely valued $f$-modules, an addendum},
url = {http://eudml.org/doc/30642},
volume = {51},
year = {2001},
}

TY - JOUR
AU - Steinberg, Stuart A.
TI - Finitely valued $f$-modules, an addendum
JO - Czechoslovak Mathematical Journal
PY - 2001
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 51
IS - 2
SP - 387
EP - 394
AB - 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.
LA - eng
KW - lattice-ordered module; value set; lattice-ordered module; value set; lattice-ordered group
UR - http://eudml.org/doc/30642
ER -