Valuations of lines
Commentationes Mathematicae Universitatis Carolinae, Tome 33 (1992) no. 4, pp. 667-679
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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.
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.
Classification :
03E70, 20M14, 20M99, 54E35
Keywords: valuation; triad; metrization theorem; semigroup
Keywords: valuation; triad; metrization theorem; semigroup
@article{CMUC_1992_33_4_a11,
author = {Ml\v{c}ek, Josef},
title = {Valuations of lines},
journal = {Commentationes Mathematicae Universitatis Carolinae},
pages = {667--679},
year = {1992},
volume = {33},
number = {4},
mrnumber = {1240188},
zbl = {0784.03033},
language = {en},
url = {http://geodesic.mathdoc.fr/item/CMUC_1992_33_4_a11/}
}
Mlček, Josef. Valuations of lines. Commentationes Mathematicae Universitatis Carolinae, Tome 33 (1992) no. 4, pp. 667-679. http://geodesic.mathdoc.fr/item/CMUC_1992_33_4_a11/
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