Decidability of General Extensional Mereology

The signature of the formal language of mereology contains only one binary predicate P which stands for the relation “being a part of”. Traditionally, P must be a partial ordering, that is, $${\forall{x}Pxx, \forall{x}\forall{y}((Pxy\land Pyx)\to x=y)}$$ and $${\forall{x}\forall{y}\forall{z}((Pxy\land Pyz)\to Pxz))}$$ are three basic mereological axioms. The best-known mereological theory is “general extensional mereology”, which is axiomatized by the three basic axioms plus the following axiom and axiom schema: (Strong Supplementation) $${\forall{x}\forall{y}(\neg Pyx\to \exists z(Pzy\land \neg Ozx))}$$, where Oxy means $${\exists z(Pzx\land Pzy)}$$, and (Fusion) $${\exists x\alpha \to \exists z\forall y(Oyz\leftrightarrow \exists x(\alpha \land Oyx))}$$, for any formula α where z and y do not occur free. In this paper, I will show that general extensional mereology is decidable, and will also point out that the decidability of the first-order approximation of the theory of complete Boolean algebras can be shown in the same way.