Abstract: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.
Keywords: valuation, triad, metrization theorem, semigroup
AMS Subject Classification: 03E70, 54E35, 20M14