On the clone of aggregation functions on bounded lattices

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.