Valuations of lines

Mlček, Josef. "Valuations of lines." Commentationes Mathematicae Universitatis Carolinae 33.4 (1992): 667-679. <http://eudml.org/doc/247393>.

@article{Mlček1992,
abstract = {We enlarge the problem of valuations of triads on so called lines. A line in an $e$-structure $\mathbb \{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 $E\circ E = \{\text\{$\mathbf \{Id\}$\}\}\upharpoonright A$) is, generally, a sequence $\mathbb \{A\}\upharpoonright B$, $\mathbb \{A\} \upharpoonright U _c$, $c\in \{\text\{$\mathbf \{FZ\}$\}\}$ (where $\{\text\{$\mathbf \{FZ\}$\}\}$ is the class of finite integers) of substructures of $\mathbb \{A\}$ such that $B\subseteq U_c \subseteq U_d$ holds for each $c\le d$. We denote this line as $\mathbb \{A\} (U_c ,B)_\{c\in \{\text\{$\mathbf \{FZ\}$\}\}\}$ and we say that a mapping $H$ is a valuation of the line $\mathbb \{A\} (U_c ,B)_\{c\in \{\text\{$\mathbf \{FZ\}$\}\}\}$ in a line $\hat\{\mathbb \{A\}\} (\hat\{U\}_c ,\hat\{B\})_\{c\in \{\text\{$\mathbf \{FZ\}$\}\}\}$ if it is, for each $c\in \{\text\{$\mathbf \{FZ\}$\}\}$, a valuation of the triad $\mathbb \{A\} (U_c,B)$ in $\hat\{\mathbb \{A\}\} (\hat\{U\}_c,\hat\{B\})$. 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.},
author = {Mlček, Josef},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {valuation; triad; metrization theorem; semigroup; Alternative Set Theory; valuations of triads; semigroup; metrization theorem},
language = {eng},
number = {4},
pages = {667-679},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Valuations of lines},
url = {http://eudml.org/doc/247393},
volume = {33},
year = {1992},
}

TY - JOUR
AU - Mlček, Josef
TI - Valuations of lines
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1992
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 33
IS - 4
SP - 667
EP - 679
AB - We enlarge the problem of valuations of triads on so called lines. A line in an $e$-structure $\mathbb {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 $E\circ E = {\text{$\mathbf {Id}$}}\upharpoonright A$) is, generally, a sequence $\mathbb {A}\upharpoonright B$, $\mathbb {A} \upharpoonright U _c$, $c\in {\text{$\mathbf {FZ}$}}$ (where ${\text{$\mathbf {FZ}$}}$ is the class of finite integers) of substructures of $\mathbb {A}$ such that $B\subseteq U_c \subseteq U_d$ holds for each $c\le d$. We denote this line as $\mathbb {A} (U_c ,B)_{c\in {\text{$\mathbf {FZ}$}}}$ and we say that a mapping $H$ is a valuation of the line $\mathbb {A} (U_c ,B)_{c\in {\text{$\mathbf {FZ}$}}}$ in a line $\hat{\mathbb {A}} (\hat{U}_c ,\hat{B})_{c\in {\text{$\mathbf {FZ}$}}}$ if it is, for each $c\in {\text{$\mathbf {FZ}$}}$, a valuation of the triad $\mathbb {A} (U_c,B)$ in $\hat{\mathbb {A}} (\hat{U}_c,\hat{B})$. 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.
LA - eng
KW - valuation; triad; metrization theorem; semigroup; Alternative Set Theory; valuations of triads; semigroup; metrization theorem
UR - http://eudml.org/doc/247393
ER -