Preassociative aggregation functions

TL;DR

This paper introduces and axiomatizes classes of preassociative aggregation functions, generalizing associativity, and connects them with established classes like Aczélian semigroups and t-norms.

Contribution

It provides the first axiomatizations of preassociative functions, extending classical associativity concepts in aggregation and fuzzy logic.

Findings

Axiomatizations of preassociative functions are developed.

Preassociativity generalizes classical associativity.

Connections with Aczélian semigroups and t-norms are established.

Abstract

The classical property of associativity is very often considered in aggregation function theory and fuzzy logic. In this paper we provide axiomatizations of various classes of preassociative functions, where preassociativity is a generalization of associativity recently introduced by the authors. These axiomatizations are based on existing characterizations of some noteworthy classes of associative operations, such as the class of Acz\'elian semigroups and the class of t-norms.