Refinement Monoids, Equidecomposability Types, and Boolean Inverse Semigroups

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The book presents a novel universal algebraic approach that connects Tarski's theory of equidecomposability types monoids with abstract measure theory and nonstable K-theory of rings. It introduces a monoid invariant known as the type monoid, explored through Boolean inverse semigroups. The author contrasts these techniques with existing topological methods, offering numerous positive results alongside counterexamples to enrich the understanding of these mathematical concepts.