Abstract Understanding the relation between different semantics in abstract argumentation is an important issue, not least since such semantics capture the basic ingredients of different approaches to nonmonotonic reasoning. The question we are interested in relates two semantics as follows: What are the necessary and sufficient conditions, such that we can decide, for any two sets of extensions, whether there exists an argumentation framework which has exactly the first extension set under one semantics, and the second extension set under the other semantics. We investigate in total nine argumentation semantics and give a nearly complete landscape of exact characterizations. As we shall argue, such results not only give an account on the independency between semantics, but might also prove useful in argumentation systems by providing guidelines for how to prune the search space.