Bell-type inequalities for quasi-copulas

We present all Bell-type inequalities regarding at most four random events of which not more than two are intersected at the same time. Reformulating these inequalities in the context of fuzzy probability calculus leads to related inequalities on commutative conjunctors. We prove that some of these inequalities hold for quasi-copulas or copulas. Additionally, for the important parametric family of Frank t-norms (which are also copulas), we identify for each of the inequalities the parameter values for which it is fulfilled, thereby completing the results of Pykacz and DʹHooghe.