I have engaged with the Lewis-style set-theoretic mereology in a few papers, undertaken jointly with Makoto Kikuchi. My interest in this topic was inspired originally by a MathOverflow question, Why hasn't mereology succeeded as a foundation of mathematics?.
Kikuchi and I had aimed to answer that question and to investigate the strength of this kind of mereology as a foundation of mathematics.
Hamkins, Joel David; Kikuchi, Makoto, Set-theoretic mereology, Log. Log. Philos. 25, No. 3, 285-308 (2016). ZBL1369.03047.
Hamkins, Joel David;
Kikuchi, Makoto, The inclusion relations of the countable models of set theory are all isomorphic, arxiv:1704.04480, 2017.
Ultimately, our conclusion is that pure set-theoretic mereology, that is, without the singleton operator, cannot succeed as a foundation of mathematics because it fulfills a decidable first order theory, and no such theory can be foundational.
I view this conclusion as rather devasting for the project of set-theoretic mereology as a foundation of mathematics.
Meanwhile, one can hope to augment mereology with other resources, and this is precisely what Lewis does in Parts of Classes, in which he makes pervasive use of the singleton operator
$$x\quad\mapsto\quad\{x\}.$$
Kikuchi and I point out, however, that set-theoretic mereology with the singleton operator is easily seen to be bi-interpretable with ordinary $\in$-based set theory. Namely, one can easily define $x\in y$ from the mereological set-theoretic inclusion (subset) relation and the singleton operator:
$$x\in y\quad\iff\quad \{x\}\subseteq y.$$
One can take this observation (which Lewis does not mention) as a strong criticism of Lewis's project, since it shows that the entire approach (set-theoretic mereology + singleton operator) can be seen as a simple translation of ordinary $\in$-based set theory into mereological+singleton langage. When theories are bi-interpretable, after all, it is reasonable to regard them as having the same semantic content.
Let me mention one of the striking results from our second paper, which shows how little set theory is observable in the pure set-theoretic mereology. Namely:
Theorem. (Hamkins, Kikuchi) The structures $\langle M,\subseteq^M\rangle$ arising as the mereological reduct of a countable model of ZFC set theory $\langle M,\in^M\rangle$ are all isomorphic.
Thus, in the pure language of set-theoretic mereology $\subseteq$, one cannot tell whether the continuum hypothesis is true or not, whether $V=L$ or not, whether there are large cardinals or not, and so on. None of the main set-theoretic points of contention is observable in set-theoretic mereology.
That proof uses the axiom of choice in the models, and this is required since the existence of amorphous sets in $\omega$-standard models is observable in the mereological reduct. But some questions remain: Do all countable $\omega$-standard models of ZF with an amorphous set have the same inclusion relation up to isomorphism?
