Systematizing semishortening conditions

We re-derive semi-shortening conditions for four-dimensional superconformal field theory with a different approach. These conditions have similar patterns that can be generalized to weaker constraints, including all those of F. Dolan and H. Osborn. In particular, for the case of $\mathcal{N} = 4$ super Yang-Mills theory, formulated in projective superspace, we find constraints for all BPS operators. We also give an example how constraints can be found from known ones. These constraints are a subset of our maximal set of semi-shortening conditions.