On the role of acyclicity in the study of rationality of fuzzy choice functions

Fuzzy choice functions obtained from preference relations have been recently used to develop automated negotiation systems. This contribution contains a theoretical study of coherence conditions in the process of creating a fuzzy choice function from a preference relation. In particular, the role of the acyclicity property of fuzzy preference relations is studied in the framework of rationality of fuzzy choice functions. Two different definitions of fuzzy acyclicity are compared. The two classical ways of constructing a fuzzy choice function from a given fuzzy preference relation are considered and properties such as acyclicity and completeness are proved to be sufficient conditions to ensure that the constructed function is a rational choice function. Special attention has been paid to the choice of the t-norm too. The results obtained are also compared to the classical results on rationality in the theory of crisp choice functions.