 The full strength of the theory can then be appreciated by considering that its models are closed under each of these functors, modulo the satisfiability of the relevant conditions. To be explicit: the condition ‘DzU’ is unsatisfiable, so U cannot have a complement. Likewise products are defined only for overlappers and differences only for pairs that leave a remainder. Otherwise, however, (55)–(60) yield perfectly well-behaved functors. Since such functors are the natural mereological analogues of the familiar set-theoretic operators, with ‘σ’ in place of set abstraction, it follows that the parthood relation axiomatized by GEM has essentially the same properties as the inclusion relation in standard set theory. More precisely, it is isomorphic to the inclusion relation restricted to the set of all non-empty subsets of a given set, which is to say a complete Boolean algebra with the zero element removed—a result that can be traced back to Tarski (1935: n. 4) and first proved in Grzegorczyk (1955:  §4).[22]