On the number of aggregation functions on finite chains as a generalization of Dedekind numbers

The aim of this paper is to show that the cardinality of the set of all n-ary aggregation functions defined on finite chains can be seen as a two-fold generalization of Dedekind numbers. One generalization arises naturally from the commonly used definition of aggregation function. The second one goes in the spirit of the former Dedekind's definition, i.e., it is shown that n-ary aggregation functions equipped with certain operations form a free algebra in a finitely generated variety over the set of n generators. & COPY; 2022 Elsevier B.V. All rights reserved.

Automated summary

The paper aims to demonstrate that the cardinality of the set of all n-ary aggregation functions defined on finite chains can be considered as a dual generalization of Dedekind numbers. The first generalization naturally arises from the commonly used definition of aggregation function. The second generalization follows in the spirit of Dedekind's original definition by showing that n-ary aggregation functions equipped with certain operations form a free algebra in a finitely generated variety over the set of n generators.