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 Pol 0 , 1 ( L ) of all 0, 1-preserving polynomials of L forms a natural subclass of the set 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 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. [ABSTRACT FROM AUTHOR]