Inverse and symmetric relators (Q1207362)

The author provides more of the basic calculus of relators --- see (the reviews of) the papers of the author [ibid. 50, 177-201 (1987; Zbl 0643.54033)] and of \textit{J. Mala}, ibid. 60, 291-297 (1992; see review 54026 above)]. This paper connects (i) the `topological ideas' defined by a relator \(\mathcal R\) (such as the limit and adherence sets of nets, the closures and interiors of sets, and open and closed sets), (ii) various modifications of \(\mathcal R\) (in particular, its uniform, proximal and topological modifications), (ii) the inverses of \(\mathcal R\) and of these modifications, and (iv) various symmetry properties of \(\mathcal R\), linked to the modifications in (ii). Throughout, the regularity/symmetry axiom \(R_ 0\) lies close at hand, and the ideas of \textit{A. S. Davis} [Am. Math. Mon. 68, 886-894 (1961; Zbl 0106.155)] and \textit{W. J. Pervin} [Math. Ann. 147, 316-317 (1962; Zbl 0101.405)] provide examples and augment the theory. (As this paper shows, relator theory can use terminology from other general approaches to uniformity. The mathematical community now needs a formal comparison with these approaches --- its advantages, its weaknesses, its ease of use... and above all, its completion theory).