Refinement monoids, equidecomposability types, and Boolean inverse semigroups
Adopting a new universal algebraic approach, the book explores and consolidates the link between \textit{A. Tarski}'s [Cardinal algebras. With an appendix, `Cardinal products of isomorphism types', by Bjarni Jónsson and Alfred Tarski. New York, London: Oxford University Press (1949; Zbl 0041.34502)] classical theory of equidecomposability type monoids, abstract measure theory (in the spirit of \textit{H. Dobbertin}'s work [Math. Ann. 265, 473--487 (1983; Zbl 0505.08005)] 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. \par More detailed, for an action of a group $G$ on a set $\Omega$, preserving a ring $\mathcal{B}$ of subsets of $\Omega$ (viz., a nonempty set of subsets of $\Omega$, closed under finite union and set difference), the commutative monoid freely generated by elements $[X]$, for $X \in \mathcal{B}$, subjected to the relations $[\emptyset] = 0$, $[gX] = [X]$ (for $g \in G)$, and $[X\sqcup Y] = [X] + [Y]$ (where $\sqcup$ denotes disjoint union), is called the monoid of $G$-equidecomposability and denoted by $Z^+ \langle {\mathcal{B}}\rangle // G$. It is well known that $Z^+ \langle \mathcal{B}\rangle // G$ is a conical refinement monoid. The author notes as an easy consequence of known results, that every countable conical refinement monoid appears as $Z^+ \langle \mathcal{B}\rangle // G$, and he develops the underlying algebraic theory, discussing in detail the quotients of refinement monoids by special sorts of congruences called $\mathbf{V}$-congruences. \par Having in mind representation problems in nonstable $K$-theory of rings and operator algebras, this naturally leads to type monoids of Boolean inverse semigroups. Observing that those monoids are identical to monoids of equidecomposability types, and formally similar to those appearing in nonstable $K$-theory of von Neumann regular rings, the author investigates various similarities and differences between those theories. \par In the process, it is proved that Boolean inverse semigroups, considered as algebras with a binary operation for the multiplication, a unary operation for the inversion, a constant for the zero and introduced in the book of two binary operations the skew difference and the skew addition form a congruence-permutable variety, thus also congruence-modular variety. The author deduces from this that they encode a large number of embedding problems of (not necessarily Boolean) inverse semigroups into involutary rings and $\mathrm{C}^*$-algebras.
scientific article; zbMATH DE number 6758718