On the clone of aggregation functions on bounded lattices

TL;DR

This paper explores aggregation functions on lattices using clone theory, showing that a finite set of at most 2n+2 functions can generate all such functions, with extensions to infinite lattices like real intervals.

Contribution

It establishes a finite generating set for aggregation functions on finite lattices and extends the approach to infinite lattices, connecting clone theory with lattice aggregation functions.

Findings

Finite lattices have a bounded set of generating aggregation functions.

Aggregation functions on infinite lattices involve limit processes.

The approach unifies finite and infinite lattice aggregation functions.

Abstract

The main aim of this paper is to study aggregation functions on lattices via clone theory approach. Observing that the aggregation functions on lattices just correspond to $0,1$-monotone clones, as the main result we show that for any finite $n$-element lattice $L$ there is a set of at most $2n+2$ aggregation functions on $L$ from which the respective clone is generated. Namely, the set of generating aggregation functions consists only of at most $n$ unary functions, at most $n$ binary functions, and lattice operations $\wedge,\vee$, and all aggregation functions of $L$ are composed of them by usual term composition. Moreover, our approach works also for infinite lattices (such as mostly considered bounded real intervals $[a,b]$), where in contrast to finite case infinite suprema and (or, equivalently, a kind of limit process) have to be considered.