1 - Triangular norms, looking back—triangle functions, looking ahead

Abstract

A matter that has received attention over the years is the question of replacing the hypotheses of Ling's theorem. Whether associativity can be replaced by power associativity is an open question, as is the question as to whether joint continuity can be replaced by continuity in each place. From generalization for its own sake, there seemed to be little reason to do so. Then so often is the case when an idea is ripe, in 1980 the crucial step—with various justifications—was taken virtually. Since the developments in the early 1980s, there has been an explosion of interest in triangular norms (t-norms), triangular conorms (t-conorms) and somewhat later copulas. In the succeeding years, there has hardly been an issue of Fuzzy Sets and Systems that did not contain an article in which t-norms were either an essential tool or the central topic. Indeed, one issue was devoted entirely to t-norms. Just as the introduction of t-norms leads to the body of results described in this chapter, the introduction of triangle functions leads to a similar array of issues.