Invariant means on Boolean inverse monoids

The classical theory of invariant means, which plays an important role in the theory of paradoxical decompositions, is based upon what are usually termed `pseudogroups'. Such pseudogroups are in fact concrete examples of the Boolean inverse monoids which give rise to etale topological groupoids...

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Bibliographic Details
Published in:arXiv.org 2015-03
Main Authors: Kudryavtseva, Ganna, Lawson, Mark V, Lenz, Daniel H, Resende, Pedro
Format: Article
Language:English
Subjects:
Boolean algebra
Invariants
Monoids
Online Access:Get full text
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Summary:The classical theory of invariant means, which plays an important role in the theory of paradoxical decompositions, is based upon what are usually termed `pseudogroups'. Such pseudogroups are in fact concrete examples of the Boolean inverse monoids which give rise to etale topological groupoids under non-commutative Stone duality. We accordingly initiate the theory of invariant means on arbitrary Boolean inverse monoids. Our main theorem is a characterization of when a Boolean inverse monoid admits an invariant mean. This generalizes the classical Tarski alternative proved, for example, by de la Harpe and Skandalis, but using different methods.
ISSN:2331-8422