## On the clone of aggregation functions on bounded lattices.(English)Zbl 1390.06006

Summary: 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 $$2 n + 2$$ 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 (or, equivalently, a kind of limit process) have to be considered.

### MSC:

 06B05 Structure theory of lattices 06A15 Galois correspondences, closure operators (in relation to ordered sets)

### Keywords:

monotone clone; monotone function; aggregation function; lattice
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### References:

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