Associativity of a class of two-place functions and its consequences for classes of triangular norms

TL;DR

This paper investigates the associativity of a class of two-place functions derived from triangular norms and strictly increasing functions, providing conditions for associativity and characterizing classes of triangular norms generated by such functions.

Contribution

It establishes that associativity depends solely on the range of the increasing function and offers necessary and sufficient conditions for associativity in specific cases.

Findings

Associativity depends only on the range of the function f.

Provides necessary and sufficient conditions for associativity when F is an ordinal sum.

Characterizes classes of triangular norms generated by strictly increasing functions.

Abstract

This article characterizes the associativity of two-place functions $T: [0,1]^2\rightarrow [0,1]$ defined by $T(x,y)=f^{(-1)}(F(f(x),f(y)))$ where $F:[0,1]^2\rightarrow[0,1]$ is a triangular norm (even a triangular subnorm), $f: [0,1]\rightarrow [0,1]$ is a strictly increasing function and $f^{(-1)}:[0,1]\rightarrow[0,1]$ is the pseudo-inverse of $f$. We prove that the associativity of functions $T$ only depends on the range of $f$, which is used to give a sufficient and necessary condition for the function $T$ being associative when the triangular norm $F$ is an ordinal sum of triangular norms and an ordinal sum of triangular subnorms in the sense of A. H. Clifford, respectively. These results finally are applied for describing classes of triangular norms generated by strictly increasing functions.