We enlarge the problem of valuations of triads on so called lines. A line in an $e$-structure $𝔸=\langle A,F,E\rangle$ (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 $E\circ E=\mathrm{𝐈𝐝}\upharpoonright A$) is, generally, a sequence $𝔸\upharpoonright B$, $𝔸\upharpoonright U_c$, $c\in \mathrm{𝐅𝐙}$ (where $\mathrm{𝐅𝐙}$ is the class of finite integers) of substructures of $𝔸$ such that $B\subseteq U_c\subseteq U_d$ holds for each $c\le d$. We denote this line as $𝔸{(U_c,B)}_{c\in \mathrm{𝐅𝐙}}$ and we say that a mapping $H$ is a valuation of the line $𝔸{(U_c,B)}_{c\in \mathrm{𝐅𝐙}}$ in a line $\widehat{𝔸}{({\widehat{U}}_c,\widehat{B})}_{c\in \mathrm{𝐅𝐙}}$ if it is, for each $c\in \mathrm{𝐅𝐙}$, a valuation of the triad $𝔸(U_c,B)$ in $\widehat{𝔸}({\widehat{U}}_c,\widehat{B})$. Some theorems on an...

This is a contribution to the theory of topological vector spaces within the framework of the alternative set theory. Using indiscernibles we will show that every infinite set $uS\subseteq G$ in a biequivalence vector space $\langle W,M,G\rangle$, such that $x-y\notin M$ for distinct $x,y\in u$, contains an infinite independent subset. Consequently, a class $X\subseteq G$ is dimensionally compact iff the $\pi$-equivalence ${\doteq }_M$ is compact on $X$. This solves a problem from the paper [NPZ 1992] by J. Náter, P. Pulmann and the second author.

Given a free ultrafilter $p$ on $\mathbb{N}$ and a space $X$, we say that $x\in X$ is the $p$-limit point of a sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ in $X$ (in symbols, $x=p$-${lim}_{n\to \infty }{x}_n$) if for every neighborhood $V$ of $x$, $\{n\in \mathbb{N}:x_n\in V\}\in p$. By using $p$-limit points from a suitable metric space, we characterize the selective ultrafilters on $\mathbb{N}$ and the $P$-points of ${\mathbb{N}}^{*}=\beta \left(\mathbb{N}\right)\setminus \mathbb{N}$. In this paper, we only consider dynamical systems $(X,f)$, where $X$ is a compact metric space. For a free ultrafilter $p$ on ${\mathbb{N}}^{*}$, the function $f^p:X\to X$ is defined by $f^p\left(x\right)=p$-${lim}_{n\to \infty }{f}^n\left(x\right)$ for each $x\in X$. These functions are not continuous in general....

The notion of dimensionally compact class in a biequivalence vector space is introduced. Similarly as the notion of compactness with respect to a $\pi$-equivalence reflects our nonability to grasp any infinite set under sharp distinction of its elements, the notion of dimensional compactness is related to the fact that we are not able to measure out any infinite set of independent parameters. A fairly natural Galois connection between equivalences on an infinite set $s$ and classes of set...