Axiomatic Considerations of t-norms and R-implications

Triangular norms (t-norms) have been applied to many areas of artificial intelligence. Residual implications (R-implications) induced by t-norms have become important tools in both theoretical and applied researches of nonclassical logics and approximate reasoning. This paper discusses axiomatic characterizations of t-norms and R-implications. The correspondences between properties of t-norms and R-implications are investigated. Some important conclusions are proved first time. Some mistakes in the related literature are corrected, and the proof procedures of some results are simplified.