The construction of left-continuous t-norms: a geometric approach in two dimensions

Each t-norm can be identified with its Cayley tomonoid, which consists of pairwise commuting order-preserving functions from the real unit interval to itself. Cayley tomonoids provide an easily manageable, yet versatile tool for the construction of t-norms. To give evidence to this claim, we review and reformulate several construction methods that are known in the literature. We adopt, on the one hand, a geometric point of view. Manipulations with t-norms have often been inspired by their three-dimensional graphs; by means of Cayley tomonoids, the process gains the character of putting together the pieces of a jigsaw puzzle. We consider, on the other hand, construction methods in their algebraic context. We show that those constructions that correspond to certain tomonoid extensions can be described on the basis of Cayley tomonoids in a particularly transparent way.