Fuzzy proximity structures and fuzzy ultrafilters (Q795358)

The investigations of fuzzy basic proximity given by \textit{K. K. Azad} [ibid. 79, 13-17 (1981; Zbl 0452.54008)] and fuzzy LO-proximity, given by the authors [ibid. 82, 491-502 (1981; Zbl 0485.54006)] are carried out. The properties of fuzzy stacks, fuzzy filters, fuzzy ultrafilters and fuzzy grills are studied. Let \((X,\Pi)\) be a fuzzy basic proximity space; then \(\Pi\) induces a fuzzy closure space \((X,c_{\alpha})\). Fuzzy \(\Pi\)- clan on X, fuzzy \(\Pi\)-cluster and fuzzy \(\Pi\)-bunch are defined. Every fuzzy \(\Pi\)-cluster is a maximal \(\Pi\)-clan; every fuzzy \(\Pi\)-cluster is a fuzzy \(\Pi\)-bunch. Some equivalences for fuzzy \(\Pi\)-clusters and fuzzy \(\Pi\)-bunches are given. A characterization of fuzzy LO-proximity in terms of fuzzy basic proximity is obtained. If \(\Pi\) is a fuzzy LO- proximity on X and \(\lambda \in \Pi(\mu)\) then there exists a fuzzy \(\Pi\)-bunch containing \(\lambda\) and \(\mu\). Some other properties are proved.