# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1812.09534