Invariant means on Boolean inverse monoids

Abstract

Abstract. 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 toétale 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.

Refinement Monoids, Equidecomposability Types, and Boolean Inverse Semigroups

“…Our additive ideals are called ∨-ideals in Kudryavtseva et al [71]. The following result shows that the concepts introduced above occur only between Boolean inverse semigroups.…”

“…In Kudryavtseva et al [71], this study is conveniently formalized in the language of inverse semigroups. Further connections can be found in works on K-theory of rings, such as Ara and Exel [7].…”

“…We also describe the congruence associated with an additive ideal. Section 3.5 is of preparatory nature, and it introduces a minor extension of the concept of generalized rook matrices introduced in Kudryavtseva et al [71]. The results of that section are applied in Section 3.6 to an extension, to the class of all Boolean inverse semigroups, of the ring-theoretical concept of crossed product.…”

“…The following concept is taken from Wallis [116, § 4.5], see also Kudryavtseva et al [71]. It extends the one of a rook matrix introduced in Solomon [103].…”

“…The type monoid Typ S is introduced in Wallis [116] and Kudryavtseva et al [71]; in the latter paper, it is denoted by T(S). In the present work, we define it in a different, though equivalent, fashion, taking advantage of the results of Chapter 2, on extending a partial refinement monoid to a full refinement monoid.…”