Remark 1: The purely mereological approach is usually understood in terms of classical extensional mereology. However, such an approach doesn’t seem to require the full power of this theory. The notion of sum used could be the join operation ∨ in a join semi-lattice. And the notion of parthood employed could be the order relation ≤ defined in this way: x ≤ y =def x ∨ y = y. The notion of join semi-lattice is more general and far less constrained than classical extensional  mereology (see Moltmann 1997, ch. 1, for other criticisms of the mereological extensional view).