Archimedean components of triangular norms

Abstract

The Archimedean components of triangular norms (which turn the closed unit interval into an abelian, totally ordered semigroup with neutral element 1) are studied, in particular their extension to triangular norms, and some construction methods for Archimedean components are given. The triangular norms which are uniquely determined by their Archimedean components are characterized. Using ordinal sums and additive generators, new types of left-continuous triangular norms are constructed.2000 Mathematics subject classification: primary 20M05; secondary 03E72.

Cancellativity properties for t-norms and t-subnorms

“…They play an important role when investigating the structure of t-norms. Under certain additional conditions they even fully determine the t-norm considered [12]. In this section we point out how each non-trivial Archimendean component determines (up to isomorphism) a unique Archimedean t-subnorm.…”

“…The Archimedean components of t-norms indicate in some sense those subintervals of ½0; 1 where the Archimedean property holds [12]. They play an important role when investigating the structure of t-norms.…”

“…( [12]) Let T be a t-subnorm. Two elements x; y 2 ½0; 1 are called Archimedean equivalent if there exists n 2 N such that x ðnÞ T 6 y 6 x or y ðnÞ T 6 x 6 y: For each x 2 ½0; 1; the equivalence class I x containing x is called an Archimedean class of T.…”

“…We call an Archimedean class trivial if j I a j¼ 1, i.e., I a ¼ fag. Each pair ðI a ; Tj I 2 a Þ is a subsemigroup of ð½0; 1; TÞ and it is called an Archimedean component of T [12]. For every a; b 2 ½0; 1 s.t.…”

H-transformation of t-norms

“…Recall also that the Archimedean components of triangular norms were recently studied in [11] and that they play an important role in the investigation of the structure of triangular norms.…”