The Study Of The Fuzzy Choice Function Based On The Fuzzy Best Element Set

In this paper, we deal with the fuzzy choice function based on the fuzzy best element set. The main research includes conditions for the fuzzy best element set to generate a fuzzy choice function, the relationships between various fuzzy preference relations derived from this choice function and the rationality properties which the choice function satisfies.Firstly, we discuss the crisp choice function based on the exact best element set briefly. While reviewing the related research about the crisp choice function based on the best element set, we also obtain some further results, such as a necessary and sufficient condition for the best element set to generate a crisp choice function, relationships between various revealed preference relations derived from this choice function and further rationality properties the choice function possesses etc.Secondly, we carry out the research on the fuzzy choice function based on the fuzzy best element set. Some conditions for the fuzzy best element set to generate a fuzzy choice function are given. Our main result is:That the fuzzy best element set can generate a fuzzy choice function and every choice set is normal iff Q satisfies strong completeness and acyclicity provided that Q is a fuzzy relation on a finite set. In view that a choice function can generate various preferences, we investigate the relationships between these various fuzzy preference relations revealed by the fuzzy choice function. Under the assumptions that every involved choice set is normal, the t-norm is left-continuous and the R-implication generated from t-norm satisfies contrapositive symmetry, we obtain a necessary and sufficient condition for various preferences to be equal. Afterwards, we discuss rationality properties of the fuzzy choice function based on the fuzzy best element set in detail. Without any further assumption, we prove that the fuzzy choice function satisfies the conditions Fα、Fγ and FCH. When the fuzzy relation involved satisfies T-transitivity, the fuzzy choice function satisfies WFCA、SFCA、Fβ+、Fβ’、 FBO etc. It is worth noticing that the fuzzy choice function based on the fuzzy best element set as a special fuzzy choice function has the unique rationality properties which are not available with a general fuzzy choice function. For example, if a fuzzy choice function is generated from the best element set corresponding to a fuzzy relation Q and every choice set is normal, then Fβ+is equivalent to T-transitivity of Q. In addition, under the assumptions that every choice set is normal and the involved t-norm is the minimum, Fβ is equivalent to transitivity of Q etc.In summary, we carry out a systematic investigation into the fuzzy choice function based on the fuzzy best element set and enrich the theory of fuzzy choice function as a result. Considering that the fuzzy best element set is based on a fuzzy preference, our research provides a reference to the investigation into other preference-based fuzzy choice functions.