One-to-one correspondences between \(\varepsilon\)-partitions, \((1-\varepsilon)\)-equivalences and \(\varepsilon\)-pseudometrics (Q1349180)

For \(\delta\) and \(\varepsilon\) in the unit interval, \(\varepsilon < \delta\), the \(\delta\)-\(\varepsilon\)-partition of a fuzzy set \(A\) is defined such that the \(\alpha\)-cuts for \(\varepsilon < \alpha < \delta\) are (classical) partitions of the respective \(\alpha\)-cut of \(A\). Such a partition is compared to the previous concepts of Ruspini, Bezdek \& Harris, Butnariu, Markechová, de Baets \& Mesiar, Chakrabort \& Das, Bhakat \& Das and Thiele. The authors concentrate on the case of \((1-\varepsilon)\)-\(\varepsilon\)-partitions. The connection of such partitions to a type of equality relations is studied, where two fuzzy sets are said to be \(\varepsilon\)-equal, if their \(\alpha\)-cuts for \(\varepsilon < \alpha < 1-\varepsilon\) are equal. It is shown that there is a one-to-one correspondence between these equivalences and partitions.