On lattices with a smallest set of aggregation functions

Given a bounded lattice $L$ with bounds $0$ and $1$, it is well known that the set ${\mathsf{P}\mathsf{o}\mathsf{l}}_{0,1}(L)$ of all $0,1$-preserving polynomials of $L$ forms a natural subclass of the set $\mathsf{C}(L)$ of aggregation functions on $L$. The main aim of this paper is to characterize all finite lattices $L$ for which these two classes coincide, i.e. when the set $\mathsf{C}(L)$ is as small as possible. These lattices are shown to be completely determined by their tolerances, also several sufficient purely lattice-theoretical conditions are presented. In particular, all simple relatively complemented lattices or simple lattices for which the join (meet) of atoms (coatoms) is $1$ ($0$) are of this kind.