We consider a set-valued mapping between two finite sets and define the alldifferent kernel which describes the submapping of alldifferent selections. This submapping is characterized by Hall partitions which are introduced in this paper. The existence of a Hall partition is equivalent to the Hall condition. The unicity of Hall partitions is proved and the unicity of an alldifferent selection is characterized. A calculation method for the determination of the Hall partition and the alldifferent kernel is presented.