One of the streams in the early development of set theory was an attempt to use mereology, a formal theory of parthood, as a foundational tool. The first such attempt is due to Stanisław Leśniewski (1886–1939). The attempt failed, but there is another, prima facie more promising attempt by Jerzy Słupecki (1904–1987), who employed his generalized mereology to build mereological foundations for type theory. I have recently written a paper explaining what generalized mereology is and why it still fails as a foundation tool. The paper is available in History and Philosophy of Logic, but if you don't have access to that, a draft is available here.