Characterizations of classes of countable Boolean inverse monoids

TL;DR

This work classifies countably infinite Boolean inverse monoids by three hierarchical regimes: finite type, AF, and UHF. It establishes that finite type corresponds to factorizable monoids with locally finite unit groups, while AF corresponds to basic monoids with locally finite units, with a constructive path from finite subalgebras to the whole. The UHF case is treated via MV-algebras, showing that S is UHF exactly when it is AF a Foulis monoid and its associated MV-algebra $ extsf{L}(S)$ is rational, enabling a bridge between inverse semigroup theory and rational MV-algebras. The results connect to non-commutative Stone duality and the KS groups, offering a pathway to classify unit groups through MV-algebraic invariants and identifying UHF monoids with Tarski-type simple monoids.

Abstract

A countably infinite Boolean inverse monoid that can be written as an increasing union of finite Boolean inverse monoids (suitably embedded) is said to be of finite type. Borrowing terminology from $C^{\ast}$-algebra theory, we say that such a Boolean inverse monoid is AF (approximately finite) if the finite Boolean inverse monoids above are isomorphic to finite direct products of finite symmetric inverse monoids, and we say that it is UHF (uniformly hyperfinite) if the finite Boolean inverse monoids are in fact isomorphic to finite symmetric inverse monoids. We characterize abstractly the Boolean inverse monoids of finite type and those which are AF and, by using MV-algebras, we also characterize the UHF monoids.

Theorems & Definitions (85)

• Remark 1.1
• Lemma 2.1
• Proposition 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• Lemma 2.6
• Proposition 2.7