  Actually one is able to axiomatize t-norm based systems for some particular classes of t-norms. As a fundamental result, Hájek (1998) has given an axiomatization of the logic BL of all continuous t-norms. Besides the previously mentioned algebraic semantics this logic has, as conjectured by Hajek and proved in Cignoli/Esteva/Godo/Torrens (2000), as another algebraic semantics the class of all t-norm based structures whose t-norm is a continuous function. Based upon this work, Esteva and Godo (2001) conjectured an axiomatization for the logic MTL of all t-norms which have the sup-preservation property, and Jenei/Montagna (2002) proved that this really is an adequate axiomatization. And Esteva/Godo/Montagna (2004) offer a method to axiomatize the logic of each single continuous t-norm:  they provide an algorithm which gives for each particular continuous t-norm \(\rT\) a finite list of axiom schemata which, if added to the logic BL of all continuous t-norms, yield an adequate axiomatization of the particular  t-norm based logic for \(\rT\).