Cyclic Evaluation of Transitivity of Reciprocal Relations

A general framework for studying the transitivity of reciprocal relations is presented. The key feature is the cyclic evaluation of transitivity: triangles (i.e. any three points) are visited in a cyclic manner. An upper bound function acting upon the ordered weights encountered provides an upper bound for the ‘sum minus 1’ of these weights. Commutative quasi-copulas allow to translate a general definition of fuzzy transitivity (when applied to reciprocal relations) elegantly into the framework of cycletransitivity. Similarly, a general notion of stochastic transitivity corresponds to a particular class of upper bound functions. Ample attention is given to self-dual upper bound functions.