# On generating of idempotent aggregation functions on finite lattices

## TL;DR

This paper characterizes the generation of idempotent aggregation functions on finite lattices using lattice operations and specific ternary functions, advancing the algebraic understanding of these functions.

## Contribution

It provides a generating set for intermediate (idempotent) aggregation functions on finite lattices, expanding the algebraic framework for their construction.

## Key findings

- All aggregation functions on finite lattices can be composed from lattice operations and certain unary and binary functions.
- A specific generating set for idempotent aggregation functions is identified, including lattice operations and ternary functions.
- The approach uses clone theory to analyze aggregation functions on lattices.

## Abstract

In a recent paper we proposed the study of aggregation functions on lattices via clone theory approach. Observing that aggregation functions on lattices just correspond to $0,1$-monotone clones, we have shown that all aggregation functions on a finite lattice $L$ can be obtained as usual composition of lattice operations $\wedge,\vee$, and certain unary and binary aggregation functions. The aim of this paper is to present a generating set for the class of intermediate (or, equivalently, idempotent) aggregation functions. This set consists of lattice operations and certain ternary idempotent aggregation functions.