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arXiv · 1812.09534

On the clone of aggregation functions on bounded lattices

Radomír Halaš·Jozef Pócs
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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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Radomír Halaš, Jozef Pócs. 2018-12-22. On the clone of aggregation functions on bounded lattices. https://doi.org/10.1016/j.ins.2015.09.038

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