Basics of a formal theory of fuzzy partitions

A theory of fuzzy partitions is an important part of any theory meant to provide a formal framework for fuzzy mathematics. In [3], Henkin-style higher-order fuzzy logic is introduced and proposed as a foundational theory for fuzzy mathematics. Here we investigate the properties of fuzzy partitions within its formal framework. We follow closely the methodology of [2]. Therefore the notions introduced here are inspired by (and deduced from) the corresponding notions of classical mathematics. Sometimes they coincide with already known notions in fuzzy literature. However, we are usually more general (we work in arbitrary fuzzy logic), more expressive (we deal with the graded properties of fuzzy relations, as in [7]), and the proofs are more elegant (resembling the classical proofs)