Relators generating the same generalized topology (Q1207376)

Relators were introduced quite recently, as a general approach to uniformity: a relator is a non-void collection of reflexive relations on a set --- see [\textit{Á. Száz}, ibid. 50, 177-201 (1987; Zbl 0643.54033)]. With each relator \(\mathcal R\) on a set \(X\), one associates its interior operator \(\text{int}_{\mathcal R}\) on \(X\) and its topological refinement \(\widehat{\mathcal R}\). According to Száz, (i) \(x\in\text{int}_{\mathcal R}(A)\) iff \({\mathcal R}(x)\subset A\) for some \(R\in{\mathcal R}\), (ii) \(A\) is an \({\mathcal R}\)-open set if \(A\subset\text{int}_{\mathcal R}(A)\), (iii) \(S\in\widehat{\mathcal R}\) iff \(x\in\text{int}_{\mathcal R}(S(x))\), for all \(x \in X\), and (iv) \(\widehat{\mathcal R}\) is the largest relator on \(X\) with the same interior operator as \(\mathcal R\). Here, the author deals with the collection of \({\mathcal R}\)-open sets, obtaining analogous results with the help of the transitive closure of \(\mathcal R\) and/or of \(\widehat{\mathcal R}\) --- in particular, the latter is the largest relator with the same open sets as \(\mathcal R\).