On The τ_T-Product of Symmetric and Subsymmetric Distribution Functions

Abstract

Let F be a non-defective distribution function (d.f.) and let $\bar F$ be the d.f. defined by $\bar F\,(x)\, = \,1\,\, - \,\,\ell ^ + F\,( - x)$ Then F is symmetric if $F \le \bar F$; and we say that F is subsymmetric if $F \le \bar F$. If T is a continuous t-norm and τ_T is the induced triangle function, then for any non-defective d.f.’s F and G we have $\tau _T (\bar F,\bar G) \le \overline {\tau _T (F,G)}$. It follows that if F and G are subsymmetric then $\tau \,_T \,(F,\,G)\,$ is subsymmetric. However, $\tau \,_T \,(F,\,G)\,$ is symmetric for all symmetric F and G only if T = Min.