Coordinatizing some concrete MV algebras and a decomposition theorem

Abstract

MV algebras are the Lindenbaum–Tarski algebras of Łukasiewicz many-valued logics. From work of D. Mundici, countable such algebras correspond to certain AF C*-algebras. M. Lawson and P. Scott gave a coordinatization theorem for them, representing any countable MV-algebra as the lattice of principal ideals of an AF Boolean inverse monoid. In this note, we give two concrete examples of such a coordinatization, one for $${\mathbb {Q}}\cap [0,1]$$ and another for the so-called Chang algebra. We also discuss Bratteli diagram techniques to further the coordinatization program.