A subset system is a means for expressing a type of (join-) completeness of posets and (join-) continuity of order-preserving maps. The authors compare subset systems, and prove, essentially, that the corresponding types of completeness coincide if the corresponding types of continuity do. They show how this relates to absolutely free completions of posets (for which a new description is also presented), and as by-product exhibit a simplified proof of the result of J. Meseguer that each subset system is equivalent to a union-complete one.