On a class of left-continuous t-norms

Roberto Cignoli, Francesc Esteva, Lluís Godo, Franco Montagna

Abstract: In this paper we study the subclass of left-continuous t-norms ∗n<math><mtext>∗</mtext><msub><mi></mi><mn>n</mn></msub></math> which are definable by an arbitrary continuous t-norm ∗<math><mtext>∗</mtext></math> and a weak (i.e. non necessarily involutive) negation n by putting x∗ny=0<math><mtext>x∗</mtext><msub><mi></mi><mn>n</mn></msub><mtext>y</mtext><mspace xmlns="true" sp="0.16" width="2px" linebreak="nobreak" is="true"></mspace><mtext>=</mtext><mspace xmlns="true" sp="0.16" width="2px" linebreak="nobreak" is="true"></mspace><mtext>0</mtext></math> if x⩽n(y), x∗ny=x∗y<math><mtext>x∗</mtext><msub><mi></mi><mn>n</mn></msub><mtext>y</mtext><mspace xmlns="true" sp="0.16" width="2px" linebreak="nobreak" is="true"></mspace><mtext>=</mtext><mspace xmlns="true" sp="0.16" width="2px" linebreak="nobreak" is="true"></mspace><mtext>x∗y</mtext></math> otherwise, thus generalizing the construction of the so-called nilpotent minimum t-norms. We provide the characterization of weak negations compatible with a given continuous t-norm and conversely which are the continuous t-norms compatible with a given weak negation function.