Semigroups and triangular norms

Erich Peter Klement, Radko Mesiar, Endre Pap

Research output: Chapter in Book/Report/Conference proceedingChapter

7 Citations (Scopus)

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.

Original languageEnglish
Title of host publicationLogical, Algebraic, Analytic and Probabilistic Aspects of Triangular Norms
PublisherElsevier
Pages63-93
Number of pages31
ISBN (Print)9780444518149
DOIs
Publication statusPublished - Dec 1 2005

ASJC Scopus subject areas

  • Computer Science(all)

Cite this

Klement, E. P., Mesiar, R., & Pap, E. (2005). Semigroups and triangular norms. In Logical, Algebraic, Analytic and Probabilistic Aspects of Triangular Norms (pp. 63-93). Elsevier. https://doi.org/10.1016/B978-044451814-9/50003-3