Adopting a new universal algebraic approach this book explores and consolidates the link
between Tarski's classical theory of equidecomposability types monoids abstract measure theory
(in the spirit of Hans Dobbertin's 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.
