| Title:
|
Valuations of lines (English) |
| Author:
|
Mlček, Josef |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
33 |
| Issue:
|
4 |
| Year:
|
1992 |
| Pages:
|
667-679 |
| . |
| Category:
|
math |
| . |
| Summary:
|
We enlarge the problem of valuations of triads on so called lines. A line in an $e$-structure $\Bbb A = \langle A,F,E\rangle$ (it means that $\langle A,F\rangle$ is a semigroup and $E$ is an automorphism or an antiautomorphism on $\langle A,F\rangle$ such that $E\circ E = {\text{$\bold {Id}$}}\restriction A$) is, generally, a sequence $\Bbb A\restriction B$, $\Bbb A \restriction U _c$, $c\in {\text{$\bold {FZ}$}}$ (where ${\text{$\bold {FZ}$}}$ is the class of finite integers) of substructures of $\Bbb A$ such that $B\subseteq U_c \subseteq U_d$ holds for each $c\leq d$. We denote this line as $\Bbb A (U_c ,B)_{c\in {\text{$\bold {FZ}$}}}$ and we say that a mapping $H$ is a valuation of the line $\Bbb A (U_c ,B)_{c\in {\text{$\bold {FZ}$}}}$ in a line $\hat{\Bbb A} (\hat{U}_c ,\hat{B})_{c\in {\text{$\bold {FZ}$}}}$ if it is, for each $c\in {\text{$\bold {FZ}$}}$, a valuation of the triad $\Bbb A (U_c,B)$ in $\hat{\Bbb A} (\hat{U}_c,\hat{B})$. Some theorems on an existence of a valuation of a given line in another one are presented and some examples concerning equivalences and ideals are discussed. A generalization of the metrization theorem is presented, too. (English) |
| Keyword:
|
valuation |
| Keyword:
|
triad |
| Keyword:
|
metrization theorem |
| Keyword:
|
semigroup |
| MSC:
|
03E70 |
| MSC:
|
20M14 |
| MSC:
|
20M99 |
| MSC:
|
54E35 |
| idZBL:
|
Zbl 0784.03033 |
| idMR:
|
MR1240188 |
| . |
| Date available:
|
2009-01-08T17:59:35Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118538 |
| . |
| Reference:
|
[Gui] Guide to Alternative set theory: (in Proc. of the $1^{st}$ Symp. Mathematics in the alternative set theory), 1989.. |
| Reference:
|
[M1] Mlček J.: Approximations of $\sigma $-classes and $\pi $-classes.Comment. Math. Univ. Carolinae 20 (1979), 669-679. MR 0555182 |
| Reference:
|
[M2] Mlček J.: Valuations of structures.Comment. Math. Univ. Carolinae (1979), 20 681-695. MR 0555183 |
| Reference:
|
[M3] Mlček J.: Monotonic valuations and valuations of triads of higher types.Comment. Math. Univ. Carolinae (1981), 22 377-398. MR 0620373 |
| Reference:
|
[V] Vopěnka P.: Mathematics in the Alternative Set Theory.TEUBNER TEXTE Leipzig (1979). MR 0581368 |
| . |
