Two Axiomatic Characterizations Of A Hesitant Fuzzy Generalization Of Rough Approximation Operators

Both rough set theory and fuzzy set theory are important mathematical tools to deal with fuzzy and uncertain knowledge,which are complementary to each other.As a generalization of classical fuzzy sets,hesitant fuzzy sets are combined with rough sets to get a new rough set containing more information,which is called hesitant fuzzy rough sets.Hesitant fuzzy rough approximation operator is the most basic concept of rough set.Therefore studying its axiomatic characterization is of great significance for a deep under standing of its mathematical structure.It is an important direction to study the axiomatic methods in rough set theory to explore the minimal axiomatic set of approximate operators.Therefore,this thesis mainly studies whether the approximation operator of hesitant fuzzy rough sets can be described by single axiom.The main work is as follows:(1)We simplify the axioms in the axiom set of hesitant fuzzy rough approximation operators into one axiom,and propose a new axiom characterization form.Firstly,the axiomatic characterization of general hesitant fuzzy rough approximation operators is given,and then the axiomatization problems of hesitant fuzzy rough approximation operators generated by serial,reflexive,symmetric,transitive and equivalent hesitant fuzzy relations are studied respectively.(2)In order to make the axiomatic set of hesitant fuzzy operators more concise,the definitions of inner product,outer product,upper and lower inverse operator operations between hesitant fuzzy sets are given.According to the above definition,the axiomatic characterization results of general hesitant fuzzy approximation operators are further simplified in the form of measurement.Meanwhile,we study the single axiom characterization of hesitant fuzzy rough approximation operators derived from serial,reflexive,symmetric and transitive hesitant fuzzy relations,respectively.(3)By using the definitions and properties of inner product,outer product,upper and lower inverse operator operations between hesitant fuzzy sets,it is further proved that the hesitant fuzzy rough approximation operators corresponding to hesitant fuzzy relations satisfying various special properties(such as seriality,reflexivity,symmetry and transitivity)can also be described by single axiom.