Abstract: Some demerits of fuzzy associative memory (FAM) based on Max-T are shown when T is any t-norm, so this type of FAM is extended into a new form. So the classed Max-T FAM where T is now a triangular norm, is the generalization of t-norm. Since a Max-T FAM can be actually a mapping from a vector space to another vector space, the storage ability of the Max-T FAM where T is a triangular norm is partly analyzed with the aid of the analyses of its value domain. Further, a new concept of concomitant implication operator of a triangular norm T is presented here. It is with such concomitant implication operator that a simple general off-line learning algorithm and a general on-line learning algorithm are proposed for a class of the Max-T FAMs based on arbitrary triangular norm T. In this case, T needs no restriction of continuity, strictly increasing, archimedean property, and so on. When T is any triangular norm, it is carefully proved that, if any given multiple pattern pairs can be reliably and perfectly stored in a Max-T FAM, then the two presented learning algorithms can easily give the maximum of all the weight matrices which can be reliably and perfectly stored in the Max-T FAM. Finally, several experiments are given to testify the effectivity of a class of Max-T FAMs and its learning algorithms.