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Foundations of the theory of vector relators. (English) Zbl 1201.54023

A nonvoid family \( \mathcal R \) of binary relations on a vector space \(X\) is called a vector relator if, for \(R \in \mathcal R\) and \(x \in X\), (i) \(R(x) = x + R(0)\), (ii) \(R(0)\) is an absorbing balanced subset of \(X\), and (iii) there exists \(S \in R\) such that \(S(0) + S(0) \subset R(0)\). These postulates imply that \( \mathcal R \) is a reflexive, symmetric, uniformly transitive, and well-chained vector relator. If \( \mathcal P \) is a nonvoid family of preseminorms on \(X\), then the collection \(R_{\mathcal P}\) of all surroundings \(B(r, p) = \{(x,y) : p(x-y) < r\}\), \(p \in \mathcal P, r > 0\), is a vector relator on \(X\).
The paper presents an in-depth study of properties of vector relators under the following headings: basic facts, balanced and convex relations, induced basic tools, fundamental properties, set-valued functions and unary operations, reflexive and topological relators, proximal and well-chained relators, transitive and filtered relators, symmetric and separating relators, bounded nets, and linearity properties. Most of the results are routine deductions from definitions and other known results.

MSC:

54E15 Uniform structures and generalizations
46A19 Other “topological” linear spaces (convergence spaces, ranked spaces, spaces with a metric taking values in an ordered structure more general than \(\mathbb{R}\), etc.)
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