We examine the degree structure of the simulation relation on the proof systems for a set L. As observed, this partial order forms a distributive lattice. A greatest element exists iff L has an optimal proof system. In case L is infinite there is no least element, and the class of proof systems for L is not presentable. As we further show the simulation order is dense. In fact any partial order can be embedded into the interval determined by two proof systems f and g such that f simulates g but g does not simulate f. Finally we obtain that for any non-optimal proof system h an infinite set of proof systems that are pairwise incomparable with respect simulation and that are also incomparable to h.