Table 1 T-norms and corresponding additive generators.

Algebraic

\(T_{A} \left( {x,y} \right) = xy\)

\(g\left( t \right) = - \log \left( t \right)\)

Einstein

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

\(g\left( t \right) = - \log \left( {\frac{2 - t}{t}} \right)\)

Hamacher

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

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

Frank

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

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