On lattices with a smallest set of aggregation functions

TL;DR

This paper characterizes finite lattices where the set of all 0,1-preserving polynomials equals the entire set of aggregation functions, identifying key lattice properties that determine this minimality.

Contribution

It provides a complete characterization of finite lattices with minimal aggregation functions, linking lattice tolerances and specific structural conditions.

Findings

Lattices where polynomials and aggregation functions coincide are fully characterized.

Simple relatively complemented lattices are among those with minimal aggregation functions.

Conditions involving joins and meets of atoms and coatoms are identified.

Abstract

Given a bounded lattice $L$ with bounds $0$ and $1$, it is well known that the set $\mathsf{Pol}_{0,1}(L)$ of all $0,1$-preserving polynomials of $L$ forms a natural subclass of the set $\mathsf{C}(L)$ of aggregation functions on $L$. The main aim of this paper is to characterize all finite lattices $L$ for which these two classes coincide, i.e. when the set $\mathsf{C}(L)$ is as small as possible. These lattices are shown to be completely determined by their tolerances, also several sufficient purely lattice-theoretical conditions are presented. In particular, all simple relatively complemented lattices or simple lattices for which the join (meet) of atoms (coatoms) is $1$ ($0$) are of this kind.