Dimensional compactness in biequivalence vector spaces

J. Náter; P. Pulmann; Pavol Zlatoš

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Displaying similar documents to “Dimensional compactness in biequivalence vector spaces”

Valuations of lines

Josef Mlček (1992)

Commentationes Mathematicae Universitatis Carolinae

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We enlarge the problem of valuations of triads on so called lines. A line in an $e$ -structure $𝔸 = 〈 A, F, E 〉$ (it means that $〈 A, F 〉$ is a semigroup and $E$ is an automorphism or an antiautomorphism on $〈 A, F 〉$ such that $E \circ E = 𝐈𝐝 ↾ A$ ) is, generally, a sequence $𝔸 ↾ B$ , $𝔸 ↾ U_{c}$ , $c \in 𝐅𝐙$ (where $𝐅𝐙$ is the class of finite integers) of substructures of $𝔸$ such that $B \subseteq U_{c} \subseteq U_{d}$ holds for each $c \leq d$ . We denote this line as $𝔸 {(U_{c}, B)}_{c \in 𝐅𝐙}$ and we say that a mapping $H$ is a valuation of the line $𝔸 {(U_{c}, B)}_{c \in 𝐅𝐙}$ in a line $\hat{𝔸} {({\hat{U}}_{c}, \hat{B})}_{c \in 𝐅𝐙}$ if it is, for each $c \in 𝐅𝐙$ , a valuation of the triad $𝔸 (U_{c}, B)$ in $\hat{𝔸} ({\hat{U}}_{c}, \hat{B})$ . Some theorems on an...

Indiscernibles and dimensional compactness

C. Ward Henson, Pavol Zlatoš (1996)

Commentationes Mathematicae Universitatis Carolinae

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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 $u S \subseteq G$ in a biequivalence vector space $〈 W, M, G 〉$ , 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 $π$ -equivalence $≐_{M}$ is compact on $X$ . This solves a problem from the paper [NPZ 1992] by J. Náter, P. Pulmann and the second author.