Semigroups and triangular norms

Abstract

Triangular norms (t-norms) turn the unit interval, equipped with the natural order, into a totally ordered semigroup with neutral element 1 and annihilator 0. This chapter discusses the algebraic notions and explores the relationship between subsemigroups, Archimedean components, and triangular norms. The chapter then explains ordinal sums allowing new semigroups to be constructed from given ones. An isomorphism between two (possibly partially ordered or totally ordered or topological) semigroups must preserve the algebraic structure and, if applicable, also the partial (linear) order and the topological structure. When talking about totally ordered semigroups, strictly increasing bijections only have been considered so far. Strictly decreasing bijections, on the other hand, lead to a concept of duality: Archimedean components are, on the one hand, essential when constructing triangular norms. On the other hand, it is important to know a rich variety of possible Archimedean components to construct new t-norms (for example, as ordinal sums of semigroups). A t-norm is continuous if and only if it is an ordinal sum of continuous Archimedean t-norms. © 2005 Elsevier B.V. All rights reserved.