Geodesic
Finitely valued $f$-modules, an addendum
Czechoslovak Mathematical Journal, Tome 51 (2001) no. 2, pp. 387-394 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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

[1] M.  Anderson and T.  Feil: Lattice-Ordered Groups. D.  Reidel, Dordrecht, 1988. | MR

[2] A.  Bigard and K.  Keimel: Sur les endomorphismes conservant les polaires d’ un groupe réticulé archimédien. Bull. Soc. Math. France 97 (1970), 81–96. | MR

[3] A.  Bigard, K.  Keimel, S.  Wolfenstein: Groupes Et Anneaux Réticulés. Springer-Verlag, Berlin, 1977. | MR

[4] G.  Birkhoff and R. S.  Pierce: Lattice-ordered rings. Am. Acad. Brasil. Ci. 28 (1956), 41–69. | MR

[5] P.  Conrad: The lattice of all convex $\ell $-subgroups of a lattice-ordered group. Czechoslovak Math. J. 15 (1965), 101–123. | MR

[6] P.  Conrad: Lattice-Ordered Groups. Tulane Lecture Notes, New Orleans, 1970. | Zbl

[7] P.  Conrad and J.  Diem: The ring of polar preserving endomorphisms of an abelian lattice-ordered group. Illinois J.  Math. 15 (1971), 222–240. | DOI | MR

[8] P.  Conrad, J.  Harvey and C.  Holland: The Hahn embedding theorem for lattice-ordered groups. Trans. Amer. Math. Soc. 108 (1963), 143–169. | DOI | MR

[9] P.  Conrad and P.  McCarthy: The structure of $f$-algebras. Math. Nachr. 58 (1973), 169–191. | DOI | MR

[10] S. A.  Steinberg: Finitely-valued $f$-modules. Pacific J.  Math. 40 (1972), 723–737. | DOI | MR | Zbl