Transitivity Bounds in Additive Fuzzy Preference Structures

Transitivity plays a crucial role in preference modeling and related fields. In this paper, we discuss this property in the general context of additive fuzzy preference structures. Of particular interest is the decomposition of a large preference relation R in its symmetric part I (indifference relation) and its asymmetric part P (strict preference relation) by means of a so-called (indifference) generator i. Given the type of transitivity of a large preference relation R (w.r.t. a conjunctor) and a generator, we establish basic lower bounds and general upper bounds on the transitivity of P and I. These bounds are due to the careful design of generic counterexamples. Moreover, we identify the situations in which these bounds are effectively reached, thereby establishing connections with interesting properties such as dominance, bisymmetry, the 1-Lipschitz property and rotation invariance