Characterization of continuous t-norms compatible with Zadehʹs probability of fuzzy events

In this work we study the concept of P Z -compatibility, which holds for those t-norms for which Zadehʹs probability of fuzzy events fulfills Kolmogorovʹs axioms. Whether a probability satisfies Kolmogorovʹs axioms or not depends almost exclusively on the t-norm and dual t-conorm used to define intersection and union, respectively, of fuzzy sets. In order to better understand fuzzy probability theory, we provide a complete characterization of the P Z -compatible continuous t-norms.