On proximal properties of proper symmetrizations of relators (Q2714375)

A relator on the set \(X\) is defined as a nonvoid family of binary relations on \(X\). If \(\mathcal R\) is a relator on \(X\), the uniform refinement \(\mathcal R^*\), proximal refinemet \(\mathcal R^\#\) and topological refinement \(\mathcal R^\land\) are NEWLINE\[NEWLINE \begin{aligned} \mathcal R^* & =\{S\subset X^2:\exists R\in \mathcal R:R\subset S\},\\ \mathcal R^\# &=\{S\subset X^2:\forall A\subset X:\exists R\in \mathcal R:R(A)\subset S(A)\}, \\ \mathcal R^\land &=\{S\subset X^2:\forall x\in X:\exists R\in \mathcal R:R(x)\subset S(x)\}, \end{aligned} NEWLINE\]NEWLINE respectively. NEWLINENEWLINENEWLINERelators are investigated from the proximal point of view. Various types of symmetries and symmetrizations are defined and the connections to other proximal properties such as filteredness and finiteness are studied. The author shows that if a relator is proximally equivalent to its proper symmetrization, then it is proximally symmetric but the converse is in general not true for relators, which are not proximally finite.