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Type Theory of Special Classes of Boolean Inverse Semigroups

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Refinement Monoids, Equidecomposability Types, and Boolean Inverse Semigroups

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2188))

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Abstract

While Theorem 4.8.9 implies that the type monoid of a Boolean inverse semigroup S can be any countable conical refinement monoid, there are situations in which the structure of S impacts greatly the one of \(\mathop{\mathrm{Typ}}\nolimits S\). A basic illustration of this is given by the class of AF inverse semigroups , introduced in Lawson and Scott [77], which is the Boolean inverse semigroup version of the class of AF C*-algebras. Another Boolean inverse semigroup version of a class of C*-algebras, which we will not consider here, is given by the Cuntz inverse monoids studied in Lawson and Scott [77, § 3].

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Notes

  1. 1.

    Those inverse semigroups are called semisimple in the first version of Lawson and Scott [77], which conflicts with the usual meaning of that word in ring theory. We chose instead to introduce semisimplicity through Definition 3.7.5.

  2. 2.

    The definition of B(G) given at the bottom of Wehrung [122, p. 272] is misformulated. Namely, since G has no least element unless it is trivial, x ∖ ⊥ does not belong to B G as a rule, so x ↦ x ∖ ⊥ does not embed D G into B G . What matters here is that the elements of B G are exactly the finite (orthogonal) joins of elements of the form b ∖ a, where (a, b) ∈ G [2]. The correct definition of B G = B(G) that ensures this is given by (5.2.2).

  3. 3.

    I believe that Lindenbaum and Tarski’s proof, as printed in [109], yields only that the partial commutative monoid B∕​​∕G satisfies the implication \(\boldsymbol{a} + 2\boldsymbol{c} =\boldsymbol{ b} +\boldsymbol{ c}\ \Rightarrow \ \boldsymbol{ a} +\boldsymbol{ c} \leq ^{+}\boldsymbol{b}\). However, by Corollary 2.7.7, this still yields the desired conclusion.

  4. 4.

    Although [109, Theorem 16.10] is stated there for Abelian G, it is mentioned on [109, p. 227] that the only consequence of abelianness that is used there is a specific (and unnamed in [109]) growth condition on group words. This condition is, of course, exponential boundedness.

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  1. Département de Mathématiques, Université de Caen Normandie, Caen, France

    Friedrich Wehrung

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Wehrung, F. (2017). Type Theory of Special Classes of Boolean Inverse Semigroups. In: Refinement Monoids, Equidecomposability Types, and Boolean Inverse Semigroups. Lecture Notes in Mathematics, vol 2188. Springer, Cham. https://doi.org/10.1007/978-3-319-61599-8_5

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