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Erschienen in:
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2017 | OriginalPaper | Buchkapitel

5. Type Theory of Special Classes of Boolean Inverse Semigroups

verfasst von : Friedrich Wehrung

Erschienen in: Refinement Monoids, Equidecomposability Types, and Boolean Inverse Semigroups

Verlag: Springer International Publishing

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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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Vorheriges Kapitel Type Monoids and V-Measures
Nächstes Kapitel Constructions Involving Involutary Semirings and Rings
Fußnoten
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
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
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
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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Metadaten
Titel
Type Theory of Special Classes of Boolean Inverse Semigroups
verfasst von
Friedrich Wehrung
Copyright-Jahr
2017
Buch
Refinement Monoids, Equidecomposability Types, and Boolean Inverse Semigroups

Print ISBN: 978-3-319-61598-1

Electronic ISBN: 978-3-319-61599-8

Copyright-Jahr: 2017

DOI
https://doi.org/10.1007/978-3-319-61599-8_5