Invariant means on Boolean inverse monoids

Ganna Kudryavtseva, Mark V. Lawson*, Daniel H. Lenz, Pedro Resende

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

11 Citations (Scopus)
56 Downloads (Pure)


The classical theory of invariant means, which plays an important rôle 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 étale 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 just such 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.

Original languageEnglish
Pages (from-to)77-101
Number of pages25
JournalSemigroup Forum
Issue number1
Early online date20 Nov 2015
Publication statusPublished - Feb 2016


  • Banach–Tarski paradox
  • Inverse semigroups
  • Pseudogroups
  • The Tarski alternative

ASJC Scopus subject areas

  • Algebra and Number Theory


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