In a previous paper, we gave a new general theory of the construction of flat operators on grey-level or multivalued images from operators on binary images. While the traditional approach was based on threshold superposition, we rely instead on threshold summation, and this allows a correct formulation for non-increasing flat operators, and also for operators with non-binary outputs. We obtained then some basic properties of flat operators, valid for both increasing and non-increasing operators. Here we pursue this work by investigating further properties of flat operators, which differ in the increasing and non-increasing cases, in particular the composition, join and meet of operators, and the commutation with contrast mappings. We study dual-ity under inversion and characterise discrete linear convolution operators as flat operators. This allows to integrate various hybrid morphological operators into our framework.