For any countable MV-algebra $A$ the Lawson-Scott coordinatization process picks some unital dimension group $(G,u)$ such that $A$ coincides with the unit interval $[0,u]$ of $(G,u)$, then picks some Bratteli diagram $B$ of $(G,u)$, and finally constructs the inverse semigroup $I(B)$ having the property that $A$ is isomorphic to the MV-algebra of principal ideals of $I(B)$. One may specialize this construction as follows:

\begin{enumerate}[{\em (i)}]

\item let $(G,u)$ be the uniquely determined unital lattice ordered abelian group corresponding to

$A$ via the categorical equivalence $\Gamma$\/ between MV-algebras and unital lattice ordered abelian groups;

\item then let $B = B(A)$ be the uniquely determined direct system of simplicial groups, all with the same unit $u$,

and unit preserving monotone homomorphisms, sitting inside $(G,u)$.

\end{enumerate}

By Marra ultrasimplicial theorem, $\lim B(A)$, $U B(A)$ and $(G,u)$ are isomorphic as unital lattice ordered abelian groups.

Via Elliott classification and its $K_0$-theoretic refinements, the AF-algebra $E(A)$ given by the direct system $B(A)$ satisfies the identity $K_(E(A)) = (G,u)$.

The Murray-von Neumann order of projections of $E(A)$ is a lattice. To illustrate the geometry of this special coordinatization process we will exemplify steps (i)-(ii) in the

all-important case $A$ when is free, i.e., (by McNaughton theorem), $A$ consists of all continuous

piecewise linear continuous functions $f:[0,1]^n\rightarrow [0,1]$, each linear piece of $f$ having integer coefficients.

Our variant of the Lawson-Scott coordinatization process draws from over 65 years of MV-algebraic theory, including the McNaughton representation of free MV-algebras (1951), Chang completeness theorem $MV = HSP([0,1])$ (1959), $\Gamma$ functor theory (1986),

and the theory of MV-algebraic Schauder bases and their underlying regular/unimodular triangulations of rational polyhedra in euclidean space (1988?).