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.

A space $X$ is if for every open cover $𝒰$ of $X,$ there exists a countably compact subset $C$ of $X$ such that $\mathrm{St}(C,𝒰)=X.$ In this paper we investigate the relations between $𝒞$-starcompact spaces and other related spaces, and also study topological properties of $𝒞$-starcompact spaces.

Let $r\ge 3$, $n\ge r$ and $\pi =(d_1,d_2,...,d_n)$ be a non-increasing sequence of nonnegative integers. If $\pi$ has a realization $G$ with vertex set $V\left(G\right)=\{v_1,v_2,...,v_n\}$ such that $d_G\left(v_i\right)=d_i$ for $i=1,2,...,n$ and $v_1{v}_2\cdots v_r{v}_1$ is a cycle of length $r$ in $G$, then $\pi$ is said to be potentially $C_r^{\text{'}\text{'}}$-graphic. In this paper, we give a characterization for $\pi$ to be potentially $C_r^{\text{'}\text{'}}$-graphic.