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This is the first book to systematically study the weak systems of mereology. In its chapters, the author critically analyzes and explains core topics related to mereology, such as parthood without antisymmetry, non-existence of the zero element, and Leśniewski's notion of class and set. The book also delves into three theories of parthood: two concern the sum existence axioms, and the third contends with transitivity of parthood. This is the first systematic analysis of systems of mereology of its kind and is suitable for students, scholars, logicians, and mathematicians who wish to further their knowledge of mereology.
Original polish publication “Podstawy teorii części” by The Nicolaus Copernicus University Press

### Chapter 1. An Introduction to the Problems of the Theory of Parthood

Abstract
In everyday speech, the expression ‘part’ is usually understood as having the sense of the expressions ‘fragment’, ‘bit’, and so forth.
Andrzej Pietruszczak

### Chapter 2. “Existentially Neutral” Theories of Parts

Abstract
Let U be an arbitrary non-empty (distributive) set of objects (a so-called universe of discourse). Let $$\sqsubset$$ be the extension of the relational concept being a part with respect to U. It is a binary relation in U which strictly partially orders the universe.
Andrzej Pietruszczak

### Chapter 3. “Existentially Involved” Theories of Parts

Abstract
Theories bearing this title are those for which at least one defining condition forces the existence of mereological sums. On Sect. 2.​6.​2 we said that a theory deserving the name ‘theory of parthood’ is one in which condition $$(\mathrm {U}_{\scriptscriptstyle \mathrel {\mathsf {sum}}})$$ is in force.
Andrzej Pietruszczak

### Chapter 4. Theories Without the Assumption of Transitivity

Abstract
We have already observed (in Sect. 1.​2) that, in the literature, the transitivity of the relation is a part of is often called into question. We cited there a number of works which take different and interesting views on the matter. In this chapter, we will introduce some general approaches which will hopefully delight both defenders and opponents of the assumed transitivity of the relation. We will introduce the concept of the local transitivity  of a given relation which will be such that every transitive relation is also locally transitive.
Andrzej Pietruszczak