Table 2 T-conorms and corresponding additive generators.

Algebraic

\(S_{A} \left( {x,y} \right) = x + y - xy\)

\(h\left( t \right) = - \log \left( {1 - t} \right)\)

Einstein

\(S_{E} \left( {x,y} \right) = \frac{x + y}{{1 + xy}}\)

\(h\left( t \right) = - \log \left( {\frac{1 + t}{{1 - t}}} \right)\)

Hamacher

\(S_{H,\gamma } \left( {x,y} \right) = \frac{{x + y - xy - \left( {1 - \gamma } \right)xy}}{{1 - \left( {1 - \gamma } \right)xy}}\)

\(h\left( t \right) = - \log \left( {\frac{{\gamma + \left( {1 - \gamma } \right)\left( {1 - t} \right)}}{1 - t}} \right),\gamma > 0\)

Frank

\(S_{F,\tau } \left( {x,y} \right) = \log_{\tau } \left( {1 + \frac{{\left( {\tau^{x} - 1} \right)\left( {\tau^{y} - 1} \right)}}{\tau - 1}} \right)\)

\(h\left( t \right) = \left\{ \begin{gathered} - \log \left( {1 - t} \right),\tau = 1 \hfill \\ - \log \left( {\frac{\tau - 1}{{\tau^{1 - t} - 1}}} \right),\tau \ne 1 \hfill \\ \end{gathered} \right.\)