5 - A survey on left-continuous t-norms and pseudo t-norms

Abstract

Left-continuity of triangular norms (t-norms) is the characteristic property to make it a residuated lattice. Nowadays, residuated lattices are subjects of intense investigation in the fields of universal algebra and non-classical logic. The recently known construction methods that result in left-continuous t-norms are discussed in this chapter. The constructions are demonstrated with examples. The question of determining uniquely either a continuous Archimedean t-norm or a left-continuous t-norm on some small subsets of the unit square is also investigated. t-norms play a crucial role in several fields of mathematics and artificial intelligence. An infinite number of left-continuous t-norms can be generated with constructions, which provides a tremendously wide spectrum of choice. Among t-norms, continuous Archimedean t-norms are the most significant in applications up to now, and in particular, the role of the Frank family of t-norms is surprisingly crucial. Ordinal sum theorems have been adapted from the field of algebra into the field of t-norms. The rotation-annihilation construction cannot be generalized to the non-commutative case.