Refinement Monoids Equidecomposability Types and Boolean Inverse Semigroups Chapter 1. Background Chapter 2. Partial commutative monoids. Chapter 3. Boolean inverse semigroups and additive semigroup homorphisms Chapter 4. Type monoids and V measures. Chapter 5. Type theory of special classes of Boolean inverse semigroups. Chapter 6. Constructions involving involutary semirings and rings. Chapter 7. discussion. Bibliography Author Index. Glossary Index Adopting a new universal algebraic approach this book explores and consolidates the link between Tarskis classical theory of equidecomposability types monoids abstract measure theory (in the spirit of Hans Dobbertins work on monoid valued measures on Boolean algebras) and the nonstable K theory of rings. This is done via the study of a monoid invariant defined on Boolean inverse semigroups called the type monoid. The new techniques contrast with the currently available topological approaches. Many positive results but also many counterexamples are provided