| Title:
|
Finitely valued $f$-modules, an addendum (English) |
| Author:
|
Steinberg, Stuart A. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
51 |
| Issue:
|
2 |
| Year:
|
2001 |
| Pages:
|
387-394 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
lattice-ordered module |
| Keyword:
|
value set |
| MSC:
|
06F15 |
| MSC:
|
06F25 |
| idZBL:
|
Zbl 0979.06010 |
| idMR:
|
MR1844318 |
| . |
| Date available:
|
2009-09-24T10:43:27Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127655 |
| . |
| Reference:
|
[1] M. Anderson and T. Feil: Lattice-Ordered Groups.D. Reidel, Dordrecht, 1988. MR 0937703 |
| Reference:
|
[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 0262137 |
| Reference:
|
[3] A. Bigard, K. Keimel, S. Wolfenstein: Groupes Et Anneaux Réticulés.Springer-Verlag, Berlin, 1977. MR 0552653 |
| Reference:
|
[4] G. Birkhoff and R. S. Pierce: Lattice-ordered rings.Am. Acad. Brasil. Ci. 28 (1956), 41–69. MR 0080099 |
| Reference:
|
[5] P. Conrad: The lattice of all convex $\ell $-subgroups of a lattice-ordered group.Czechoslovak Math. J. 15 (1965), 101–123. MR 0173716 |
| Reference:
|
[6] P. Conrad: Lattice-Ordered Groups.Tulane Lecture Notes, New Orleans, 1970. Zbl 0258.06011 |
| Reference:
|
[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. MR 0285462, 10.1215/ijm/1256052710 |
| Reference:
|
[8] P. Conrad, J. Harvey and C. Holland: The Hahn embedding theorem for lattice-ordered groups.Trans. Amer. Math. Soc. 108 (1963), 143–169. MR 0151534, 10.1090/S0002-9947-1963-0151534-0 |
| Reference:
|
[9] P. Conrad and P. McCarthy: The structure of $f$-algebras.Math. Nachr. 58 (1973), 169–191. MR 0330000, 10.1002/mana.19730580111 |
| Reference:
|
[10] S. A. Steinberg: Finitely-valued $f$-modules.Pacific J. Math. 40 (1972), 723–737. Zbl 0218.16008, MR 0306078, 10.2140/pjm.1972.40.723 |
| . |
