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valuation; triad; metrization theorem; semigroup
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.
[Gui] Guide to Alternative set theory: (in Proc. of the $1^{st}$ Symp. Mathematics in the alternative set theory), 1989.
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[M2] Mlček J.: Valuations of structures. Comment. Math. Univ. Carolinae (1979), 20 681-695. MR 0555183
[M3] Mlček J.: Monotonic valuations and valuations of triads of higher types. Comment. Math. Univ. Carolinae (1981), 22 377-398. MR 0620373
[V] Vopěnka P.: Mathematics in the Alternative Set Theory. TEUBNER TEXTE Leipzig (1979). MR 0581368
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