Fuzzy Partitions And The Induced Fuzzy Rough Approximation Operators

Fuzzy set theory is a mathematical tool for dealing with uncertainties and fuzzy problems by means of set theory in the real world.Besides probability theory,fuzzy set theory and evidence theory,rough set theory is another mathematical tool for dealing with uncertainty.It can be used to analyze imprecise,inconsistent,incomplete various information.Meanwhile,it also can be used to discover tacit knowledge and potential rules behind information.The combination of fuzzy sets and rough sets is always a research hotspot in the research community.In this paper,we mainly consider the concept of fuzzy partitions and their one-to-one correspondence with fuzzy equivalences.We define a pair of fuzzy rough approximation operators based on every fuzzy partition,and study some basic properties of the fuzzy rough approximation operators.The methods and techniques of specific problems can be applied to study the covering/neighborhood-based fuzzy rough sets.The main contents of this paper are as follows:Content One: A definition of fuzzy partitions is introduced by taking the ordered monoid(without commutative property)as the truth value table.We show that this kind of fuzzy partitions is one-to-one corresponding to fuzzy equivalences.When the truth value table is a residual lattice,this definition is equivalent to that in the sense of Belohlavek.Content Two: On commutative unital quantale as the truth value table,we define the upper and lower rough approximation operators by fuzzy partitions,and then we give two equivalent descriptions of fuzzy upper and lower approximation operators.Content Three: We systematically investigate properties of upper and lower rough approximation operators.Although those properties can be obtained according to the equivalences in Content One,the specific methods and techniques can be applied to the study covering and neighborhood fuzzy rough set theory.