Refinement monoids, equidecomposability types, and Boolean inverse semigroups
type: Item
instance of: scholarly article
title: Refinement monoids, equidecomposability types, and Boolean inverse semigroups
zbMATH Open document ID: 1447.20005
DOI: 10.1007/978-3-319-61599-8
author: Friedrich Wehrung
published in: Lecture Notes in Mathematics
publication date: 2017-08-08T00:00:00Z
review text: 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 identi…
Mathematics Subject Classification ID: 20-02
Mathematics Subject Classification ID: 20M10
Mathematics Subject Classification ID: 20M32
zbMATH DE Number: 6758718
zbMATH Keywords: antigroup
zbMATH Keywords: semigroup
zbMATH Keywords: monoid
zbMATH Keywords: commutative monoid
zbMATH Keywords: inverse monoid
zbMATH Keywords: distributive lattice
zbMATH Keywords: Boolean ring
zbMATH Keywords: bias
zbMATH Keywords: tight map
zbMATH Keywords: tight enveloping algebra
zbMATH Keywords: refinement monoid
zbMATH Keywords: partial monoid
zbMATH Keywords: enveloping monoid
zbMATH Keywords: type monoid
zbMATH Keywords: equidecomposable
zbMATH Keywords: $\mathbf{V}$-measure
zbMATH Keywords: $\mathbf{V}$-homomorphism
zbMATH Keywords: $\mathbf{V}$-congruence
zbMATH Keywords: group-induced measure
zbMATH Keywords: groupoid-induced measure
zbMATH Keywords: group-measurable monoid
zbMATH Keywords: groupoid-measurable monoid
zbMATH Keywords: local chartability
zbMATH Keywords: approximately finite semigroup
zbMATH Keywords: semisimple semigroup
zbMATH Keywords: lattice-ordered semigroup
zbMATH Keywords: path algebra
zbMATH Keywords: strongly separative monoid
zbMATH Keywords: fork
reviewed by: Leonid M. Martynov
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-319-61599-8
OpenAlex ID: W2755091354
description: scientific article; zbMATH DE number 6758718