Characterizations of classes of countable Boolean inverse monoids

TL;DR

This paper characterizes classes of countable Boolean inverse monoids, including AF and UHF types, using algebraic and MV-algebraic methods, extending concepts from $C^{ ext{*}}$-algebra theory.

Contribution

It provides abstract characterizations of finite type, AF, and UHF Boolean inverse monoids, linking them to well-known algebraic structures.

Findings

Characterization of Boolean inverse monoids of finite type.

Identification of AF monoids as those built from finite symmetric inverse monoids.

Use of MV-algebras to characterize UHF 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.