The Category of Ordered Bratteli Diagrams
Massoud Amini
George A. Elliott
and Nasser Golestani
A category structure for ordered Bratteli diagrams is proposed such that isomorphism in this category coincides with Herman, Putnam, and Skau's notion of equivalence. It is shown that the one-to-one correspondence between the category of essentially minimal totally disconnected dynamical systems and the category of essentially simple ordered Bratteli diagrams at the level of objects is in fact an equivalence of categories. In particular, we show that the category of Cantor minimal systems is equivalent to the category of properly ordered Bratteli diagrams. We obtain a model (diagram) for a homomorphism between essentially minimal totally disconnected dynamical systems, which may be useful in the study of factors and extensions of such systems. Various functors between the categories of essentially minimal totally disconnected dynamical systems, ordered Bratteli diagrams, AF~algebras, and dimension groups are constructed and used to study the relations between these categories.
1509.07246.tex
\documentclass[11pt]{amsart}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Packages
\usepackage{amsmath,amsfonts,amsthm}
\usepackage{amssymb,latexsym}
%\usepackage{showkeys}
%\usepackage[colorlinks]{hyperref}
\usepackage[colorlinks=true, linkcolor=blue,
urlcolor=blue]{hyperref}
\usepackage{graphicx}
\usepackage[all]{xy}
\usepackage{layout}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% New commands
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}

\theoremstyle{definition}
\newtheorem{comment}{Comment}[section]
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{counterexample}[theorem]{Counterexample}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{note}[theorem]{Note}

\theoremstyle{remark}
\newtheorem*{notation}{Notation}
\newtheorem*{remark}{Remark}
\numberwithin{equation}{section}

\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\Prime}{\mathbb{P}}
\newcommand{\SP}{\: \: \: \: \:}
%%%%%%%%%%%%%%%%%%%%%%%%%%% Article information
\title{The Category of Ordered Bratteli Diagrams}
%\date{\today}

%%%%%%%%%%%%%%%%%%%%%%%%%%% Author information
\author{Massoud Amini,  George A. Elliott, and Nasser Golestani}

\address[\textbf{Massoud Amini}]{Department of Pure Mathematics\\ Faculty of Mathematical Sciences\\
  Tarbiat Modares University\\
 Tehran\\ Iran}
 \address{School of Mathematics, Institute for Research in Fundamental Sciences (IPM),
Tehran 19395-5746, Iran}
\email {mamini@modares.ac.ir, mamini@ipm.ir}

\address[\textbf{George A. Elliott}]{Department of Mathematics\\ University of Toronto\\ Toronto, Ontario, Canada\ \ M5S 2E4}
\email {elliott@math.toronto.edu}

\address[\textbf{Nasser Golestani}]{Department of Pure Mathematics\\ Faculty of Mathematical Sciences\\
   Tarbiat Modares University\\
 Tehran\\ Iran}
 \address{School of Mathematics, Institute for Research in Fundamental Sciences (IPM),
Tehran 19395-5746, Iran}
\email{n.golestani@modares.ac.ir}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% AMS information
\subjclass[2010]{46L05, 37B05, 37A20} \keywords{dynamical system,
ordered Bratteli diagram,
category, functor, AF~algebra, dimension group}
\thanks{The first and third authors were partly supported by a grant from IPM (94430215)}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Document
\begin{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Abstract
\begin{abstract}
A category structure for ordered Bratteli diagrams is proposed such that
isomorphism in this category coincides with Herman, Putnam, and Skau's notion
of equivalence. It is shown that the one-to-one correspondence between
the category of essentially minimal totally disconnected dynamical systems
and the category of essentially simple ordered Bratteli diagrams at the level of objects  is in fact an equivalence
of categories.
In particular,
we show that the category of Cantor minimal  systems is equivalent to the category of
properly ordered Bratteli diagrams.
We  obtain  a model (diagram) for a homomorphism between essentially minimal totally disconnected dynamical
systems, which may be useful  in the study of factors and extensions of such systems.
Various functors between the categories of
essentially minimal totally disconnected dynamical systems, ordered Bratteli diagrams, AF~algebras, and dimension groups
are constructed and used to study the relations between these categories.
\end{abstract}
\maketitle
\tableofcontents
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Introduction
\section{Introduction}
\noindent
In 1972, Bratteli in a seminal paper introduced  what are
now called Bratteli diagrams to study AF~algebras \cite{br72}.
He  associated to each AF algebra an infinite
directed graph,  its Bratteli diagram (see Definition~\ref{defbd2}),
and used these very effectively to study AF algebras.
Some attributes of an AF algebra can be read directly from its Bratteli diagram (e.g.,
 ideal structure).

 Based on the notion of a Bratteli diagram,  Elliott introduced dimension groups and gave a
  classification of AF~algebras using K-theory in 1976 \cite{el76}.
 In fact, Elliott showed that the functor
 $\mathrm{K}_{0}: \mathbf{AF}\to \mathbf{DG}$, from the category of
 AF~algebras with  $*$-homomorphisms
 to the category of dimension groups with order-preserving homomorphisms,
is a strong classification functor
 (see \cite{el76}  and \cite[Sections 5.1--5.3]{el10}).
Recall that a functor $F:\mathcal{C}\to \mathcal{D}$
was called in \cite{el10} a {classification functor} if
$F(a)\cong F(b)$ implies $a\cong b$, for each $a,b\in \mathcal{C}$,
and a {strong classification functor} if each isomorphism from $F(a)$ to $F(b)$
is the image of an isomorphism from $a$ to $b$.

In \cite{aeg14}, the authors introduced a notion of  morphism between Bratteli diagrams
such that, in the resulting category of Bratteli diagrams, $\mathbf{BD}$,
isomorphism of Bratteli diagrams coincides
with Bratteli's notion of equivalence.
We showed that the map $\mathcal{B}:\mathbf{AF}\to \mathbf{BD}$,
defined by Bratteli  in \cite{br72} on objects, is in fact a functor. The fact that this is a
strong classification functor, \cite[Theorem~3.11]{aeg14}, is a functorial formulation of Bratteli's classification
of AF~algebras and completes his work from the classification point of view
introduced by Elliott in \cite{el10}.

In a notable development, Bratteli diagrams have been used to study certain dynamical systems.
In 1981, A.\,V. Ver\v{s}ik used Bratteli diagrams to construct so-called adic transformations \cite{ve81a, ve81b}.
Based on his work (and the work of S.\,C.~Power \cite{po91}),
Herman, Putnam, and Skau introduced the notion of an ordered Bratteli diagram
and associated a dynamical system to each (essentially simple) ordered Bratteli diagram  \cite{hps92}.
In fact, they showed that there is a one-to-one correspondence between
essentially simple ordered Bratteli diagrams
and essentially minimal totally disconnected
dynamical systems \cite[Theorem~4.7]{hps92}. (In particular, each Cantor minimal system has a
Bratteli-Ver\v{s}ik model.)
This correspondence was used effectively to study
Cantor minimal dynamical systems and the characterization of various types
of orbit equivalence in terms of isomorphism of the related C*-algebra crossed products and
dimension groups \cite{hps92, gps95, gw95, ma02}.

In this paper,
we propose a notion of (ordered) morphism between ordered Bratteli diagrams and
obtain the \emph{category} of ordered Bratteli diagrams, $\mathbf{OBD}$
(Theorem~\ref{thrcatobd}). Isomorphism in this category coincides with
the notion of equivalence in the sense of Herman, Putnam, and Skau (Theorem~\ref{thrisoo}).
We show that the correspondence obtained by Herman, Putnam, and Skau in \cite{hps92}
(mentioned above)
is an equivalence of categories. Denote by
$\mathbf{SDS}$ the category of scaled essentially minimal totally disconnected dynamical systems
(see Definition~\ref{defSDS}). We construct a contravariant functor
$\mathcal{P}: \mathbf{SDS}\to\mathbf{OBD}$, thus  obtaining what might be
viewed as a model (diagram) for a homomorphism between essentially minimal totally disconnected dynamical
systems. (In particular, this may be   useful in the study of factors and extensions of such systems.)

In Section~4, we show that the contravariant functor $\mathcal{P}: \mathbf{SDS}\to\mathbf{OBD}$
is full  and faithful. The fact that this functor is full
is a tool to obtain homomorphisms between the dynamical systems in question
by graphically constructing certain arrows (i.e., morphisms) between the associated
Bratteli diagrams.
The (essential) range of this functor is the class  of
essentially simple ordered Bratteli diagrams
$\mathbf{OBD}_{\mathrm{es}}$. Hence
  $\mathcal{P}: \mathbf{SDS}\to\mathbf{OBD}_{\mathrm{es}}$
is an equivalence of categories (Theorem~\ref{threqucat}).


In Section~5, we construct a functor inverse to the functor $\mathcal{P}$, a
 contravariant functor
$\mathcal{V} : \mathbf{OBD}_{\mathrm{es}}\to \mathbf{SDS}$ which is
also an equivalence of categories. Our definition of this functor on objects
uses ideas from \cite{hps92}. We compute the correspondences
$\tau$ and $\sigma$ implementing the natural isomorphisms
$1_{\mathbf{OBD}_{\mathrm{es}}}\cong \mathcal{P}\mathcal{V}$ and
$1_{\mathbf{SDS}}\cong \mathcal{V}\mathcal{P}$.
(This is in particular useful when one wants to apply these
functors to morphisms.) In this way, we obtain a functorial formulation of the correspondence of \cite{hps92}
between essentially simple ordered Bratteli diagrams and
essentially minimal totally disconnected dynamical systems (Theorem~\ref{theequcat2}).

In particular, in Section~6 we show that the category of minimal dynamical systems based
on
compact, totally disconnected metrizable spaces is equivalent to the category of
simple ordered  Bratteli diagrams (Corollary~\ref{cormds}),
and
the category of minimal dynamical systems
on the Cantor set is equivalent to the category of
properly ordered Bratteli diagrams  (Corollary~\ref{cormdsc}).

The relations between AF~algebras, Bratteli diagrams, dimension groups, and dynamical
systems have been studied in various papers (for example, see \cite{el76, pu89, hps92, gps95, dhs99, aeg14}).
Most of the results concentrate on objects. We wish to give a functorial picture of
relations between these categories (to encompass also morphisms). Thus,
in  this note we construct the missing functors in the following commutative (up to natural isomorphism)
 diagram:

\[
\xymatrix{
& \mathbf{SDS} \ar@{^{(}->}[ld]\ar@{-->}@/^1pc/[rr]^{\mathcal{P}}
 &
 & \mathbf{OBD}_{\mathrm{es}}\ar@{_{(}->}[rd] \ar@{-->}@/^1pc/[ll]^{\mathcal{V}}& &\\
\mathbf{DS}\ar@{-->}[rr]^{\mathcal{AF}}
\ar@{-->}[rd]_{\mathrm{K}^{0}} &
& \mathbf{AF}\ar[rr]^{\mathcal{B}}\ar[ld]^{\mathrm{K}_{0}}
\ar[rd]_{\overline{\mathcal{F}}} &
& \mathbf{BD}\\
&  \mathbf{DG} & & \overline{\mathbf{AF}^{\mathrm{out}}}
\ar[ll]^{\overline{\mathrm{K}}_{0}}\ar[ru]_{\overline{\mathcal{B}}}
& &
}
\]

Here, $\mathbf{DS}$, $\mathbf{SDS}$, $\mathbf{DG}$, $\mathbf{BD}$, $\mathbf{OBD}$,
 $\mathbf{AF}$, and $\overline{\mathbf{AF}^{\mathrm{out}}}$ denote
the categories of essentially minimal totally disconnected dynamical systems,
scaled essentially minimal totally disconnected dynamical systems,
dimension groups,
Bratteli diagrams, ordered Bratteli diagrams,  AF~algebras, and the abstract category
associated with AF~algebras in \cite{el10}, respectively.
The precise definitions
of these categories will be given below.
(See \cite{aeg14} for the precise definition of the categories $\mathbf{DG}$, $\mathbf{BD}$, $\mathbf{AF}$, and
$ \overline{\mathbf{AF}^{\mathrm{out}}}$; the others are defined in the sequel.)

The functor $\mathrm{K}_{0}: \mathbf{AF}\to \mathbf{DG}$ was introduced by Elliott in \cite{el76}
and is a strong classification functor (see also \cite{el10, aeg14}). The functors
$\mathcal{B}:\mathbf{AF}\to \mathbf{BD}$   and
$\overline{\mathcal{F}} :\mathbf{AF}\to  \overline{\mathbf{AF}^{\mathrm{out}}}$
are also strong classification functors
\cite{el10, aeg14}.
The functors
$\overline{\mathrm{K}}_{0}:\overline{\mathbf{AF}^{\mathrm{out}}} \to \mathbf{DG}$ and
$\overline{\mathcal{B}}:\overline{\mathbf{AF}^{\mathrm{out}}} \to \mathbf{BD}$
are equivalence of categories \cite{aeg14}.
The functors
$\mathcal{P}: \mathbf{SDS}\to\mathbf{OBD}_{\mathrm{es}}$ and
$\mathcal{V} : \mathbf{OBD}_{\mathrm{es}}\to \mathbf{SDS}$
are also  equivalence of categories (see Sections~4 and 5).
The functor $\mathrm{K}^{0}:\mathbf{DS}\to \mathbf{DG}$ is defined in Section~8
and the functor $\mathcal{AF}:\mathbf{DS}\to \mathbf{AF}$
 is defined in Section~9 and is a faithful functor.
The direct limit functor $\mathcal{D}:\mathbf{BD}\to \mathbf{DG}$ (see Section~7)
 is an equivalence of categories.

\section{Preliminaries}
\noindent
There are various definitions of a  Bratteli diagram which are essentially equivalent.
The present work is a continuation of \cite{aeg14}, using certain ideas from
\cite{pu89, hps92, gps95}. For the definition of a  Bratteli diagram  we shall follow \cite{hps92},
but the relation to the notions of \cite{aeg14} will be discussed.
\begin{definition}[\cite{hps92}, Definition~2.1]\label{defbd2}
A Bratteli diagram consists of a vertex set $V$ and an edge set $E$ satisfying
the following conditions. We have a decomposition of $V$ as a disjoint union
$V_{0}\cup V_{1}\cup \cdots$, where each $V_{n}$ is finite and non-empty
and $V_{0}$ has exactly one element, $v_{0}$. Similarly, $E$ decomposes as
a disjoint union $E_{1}\cup E_{2}\cup \cdots$, where each $E_{n}$ is finite
and non-empty. Moreover, we have maps $r,s:E\to V$ such that $r(E_{n})\subseteq V_{n}$
and $s(E_{n})\subseteq V_{n-1}$, $n=1,2,3,\ldots$ ($r=$ range, $s=$ source).
We also assume that $s^{-1}\{v\}$ is non-empty for all $v$ in $V$ and
$r^{-1}\{v\}$ is non-empty for all $v$ in $V\setminus V_{0}$.
 Let us denote such a $B$ by the  diagram

\[
\xymatrix{V_{0}\ar[r]^-{\resizebox{1.15em}{.65em}{$E_{1}$}}
 &V_{1}\ar[r]^-{\resizebox{1.15em}{.65em}{$E_{2}$}}
 &V_{2}\ar[r]^-{\resizebox{1.15em}{.65em}{$E_{3}$}}
 &\cdots\ .
 }
\]
\end{definition}
Let us denote by $\mathbf{BD}_{1}$ the class of all Bratteli diagrams in the sense
of Definition~\ref{defbd2} (the superscript 1 will also appear later).
\begin{definition}[\cite{aeg14}, Definition~2.2]\label{defbd}
By a Bratteli diagram let us mean an ordered pair $B=(V,E)$,
$V=(V_{n})_{n=1}^{\infty}$ and $E=(E_{n})_{n=1}^{\infty}$, such that:
\begin{itemize}
\item[(1)]
each $V_{n}$ is a $k_{n}\times 1$ matrix of non-zero positive integers for
some
$k_{n}\geq 1$;
\item[(2)]
each $E_{n}$ is an embedding matrix  from $V_{n}$ to $V_{n+1}$, i.e., a
$k_{n+1}\times k_{n}$ matrix of positive integers (including zero)
with no zero columns, such that
$E_{n}V_{n}\leq V_{n+1}$, in the coordinate-wise order.
\end{itemize}
\end{definition}
As in \cite{aeg14}, we shall use the notation $\mathbf{BD}$ for the set of Bratteli diagrams
in the sense of Definition~\ref{defbd}. Denote by $\mathbf{BD}^{1}$
the set of all $(V,E)$ in $\mathbf{BD}$ such that
each $E_{n}$ has no zero column and that $E_{n}V_{n}=V_{n+1}$, for
each $n\geq 1$. In fact,  $\mathbf{BD}^{1}$ is the image of $\mathbf{AF}_{1}$,	  the class of
all unital AF~algebras,  under
the functor $\mathcal{B}:\mathbf{AF}\to \mathbf{BD}$ introduced in \cite{aeg14}
(by  \cite[Proposition~4.3]{aeg14}).

The essential difference between Definition~\ref{defbd2} and Definition~\ref{defbd}
is the specification of a total ordering on the vertices of each level.
In fact, the categories $\mathbf{BD}_{1}$ and $\mathbf{BD}^{1}$ are equivalent
(see Theorem~\ref{threqubd} below).
We will adopt Definition~\ref{defbd2} in the sequel.

Let us recall some definitions and  fix some notation concerning
 Bratteli diagrams \cite{hps92, gps95,  dhs99, aeg14}.

With the notation of Definition~\ref{defbd2}, if we fix a total order on
each $V_{n}$ then to each edge set $E_{n}$  is associated a matrix
called the  \emph{multiplicity matrix} of $E_{n}$,  which we denote  by
$\mathrm{M}(E_{n})$ (it was called the ``incidence matrix'' in \cite{gps95}; the expression ``multiplicity
matrix'' was used in \cite{aeg14} for the matrix associated to a homomorphism
between finite dimensional C$^*$-algebras).
The number of columns and rows of  $\mathrm{M}(E_{n})$ equals the number of elements
of $V_{n-1}$ and $V_{n}$, respectively, and each entry of the matrix gives the number of edges between
 two vertices.

Let $l,k$ be integers with $0\leq k< l$. Let $E_{k+1}\circ E_{k}\circ\cdots\circ E_{l}$, or also
$E_{k,l}$,
denote the set of all paths from $V_{k}$ to $V_{l}$, i.e., the following set:
\begin{equation*}
\{(e_{k+1},\ldots,e_{l}) \mid e_{i}\in E_{i}, i=k+1,\ldots,l,\  r(e_{i})=s(e_{i+1}),
i=k+1,\ldots,l\}.
\end{equation*}
We define $r(e_{k+1},\ldots,e_{l})=r(e_{l})$ and
$s(e_{k+1},\ldots,e_{l})=s(e_{k+1})$.
Note that
$\mathrm{M}(E_{k,l})= \mathrm{M}(E_{l})\cdots \mathrm{M}(E_{k})\mathrm{M}(E_{k+1})$.
Also set $E_{k,k}=\{(v,v)\mid v\in V_{k}\}$ which is an edge set from $V_{k}$ to itself
and its multiplicity matrix is the identity matrix of order equal to the cardinality of $V_{n}$.
\begin{definition}[\cite{hps92}, Definition~2.2]\label{defcontr}
Given a Bratteli diagram $(V,E)$ and a sequence $m_{0}=0<m_{1}<m_{2}<\cdots$
in $\N$, the \emph{contraction} (\textit{telescoping}) of $(V,E)$ to $\{m_{n}\}_{0}^{\infty}$ is
the Bratteli diagram $(V',E')$ where $V_{n}'=V_{m_{n}}$, for $n\geq 0$, and
$E_{n}'=E_{m_{n-1},m_{n}}$, and $r'=r$, $s'=s$ as above.
\end{definition}
Two Bratteli diagrams $(V,E)$ and $(V',E')$ were called \textit{isomorphic} in \cite{hps92}
if there exist  bijections between $V$ and $V'$
and between $E$ and $E'$, preserving the gradings and intertwining the
respective source and range maps. Also, Herman, Putnam, and Skau considered a
 notion of \textit{equivalence} of Bratteli diagrams, denoted by $\sim$, in \cite{hps92}
(basically due to Bratteli in \cite{br72}).
In fact, this is the equivalence relation
generated by isomorphism (as in \cite{hps92}) and telescoping. As indicated in \cite{dhs99},
the Bratteli diagrams $(V^{1}, E^{1})$ and $(V^{2}, E^{2})$ are equivalent
if and only if there exists a Bratteli diagram $(V,E)$ such that the telescoping $(V,E)$
to odd levels $0<1<3<\cdots$ yields a telescoping of
$(V^{1}, E^{1})$
and the telescoping of $(V,E)$ to even levels $0<2<4<\cdots$ yields a telescoping of $(V^{2}, E^{2})$.
(This relation can be seen immediately to be an equivalence relation.)

Let us recall the definition of the category of Bratteli diagrams $\mathbf{BD}_{1}$,
as introduced in \cite{aeg14},
in a way  suited to Definition~\ref{defbd2}.
As we shall see, the notion of isomorphism of Herman, Putnam, and Skau
coincides with isomorphism in the category $\mathbf{BD}_{1}$
with premorphisms (Proposition~\ref{propisopre}),
and their notion of equivalence coincides with isomorphism
in the category $\mathbf{BD}_{1}$ with morphisms (Theorem~\ref{thrisomor}).
\begin{definition}[cf.~\cite{aeg14}, Definition~2.3]\label{defprebd}
Let $B=(V,E)$ and $C=(W,S)$
be in $\mathbf{BD}_{1}$.
A \emph{premorphism} $f:B\to C$
is an ordered pair $(F, (f_{n})_{n=0}^{\infty})$ where
$(f_{n})_{n=0}^{\infty}$ is a cofinal sequence of  positive integers with
$f_{0}=0\leq f_{1}\leq f_{2}\leq\cdots$ and $F$ consists of a disjoint union
 $F_{0}\cup F_{1}\cup F_{2}\cup\cdots$ together with a pair of range and source maps
 $r:F\to W$, $s:F\to V$ such that:

 (1)
 each $F_{n}$ is a non-empty finite set, $s(F_{n})\subseteq V_{n}$,
 $r(F_{n})\subseteq W_{f_{n}}$,  $F_{0}$
 is a singleton,
 $s^{-1}\{v\}$ is non-empty for all $v$ in $V$, and
$r^{-1}\{w\}$ is non-empty for all $w$ in $W$;

(2)
 the diagram of $f:B\to C$  commutes:

 \[
\xymatrix{V_{0}\ar[r]^{E_{1}}\ar[d]_{F_{0}}
 &V_{1}\ar[r]^-{E_{2}}\ar[d]_{F_{1}} &V_{2}\ar[r]^-{E_{3}}\ar[ld]^{F_{2}} &\cdots \ \ \  \\
 W_{0}\ar[r]_{S_{1}}
 &W_{1}\ar[r]_{S_{2}} &W_{2}\ar[r]_{S_{3}}&\cdots \   .
 }
\]
By commutativity of the diagram, we mean that
for each $n\geq 0$, each $v\in V_{n}$, and each $w\in W_{f_{n+1}}$,
 the number of paths from $v$ to $w$ passing through $W_{f_{n}}$ equals
 that of ones passing through $V_{n+1}$;
 we use the notation
 $E_{n}\circ F_{n+1}\cong F_{n}\circ S_{f_{n},f_{n+1}}$ to refer to this fact.
\end{definition}
\begin{remark}
With the notation of Definition~\ref{defprebd},
 the diagram of $f:B\to C$  commutes if, and only if, for any positive integer $n$
  we have  $\mathrm{M}(F_{n+1})\mathrm{M}(E_{n})=\mathrm{M}(S_{f_{n},f_{n+1}})\mathrm{M}(F_{n})$,
  i.e., the square

\[
\xymatrix{V_{n}\ar[rr]^{\mathrm{M}(E_{n})}\ar[d]_{\mathrm{M}(F_{n})}
 & &V_{n+1}\ar[d]^{\mathrm{M}(F_{n+1})}  \\
 W_{f_{n}}\ar[rr]_{\mathrm{M}(S_{f_{n},f_{n+1}})} &
 & W_{f_{n+1}}
 }
\]

\noindent commutes. (This implies the general property of commutativity, namely, that any two paths of maps between the same pair of points in the diagram agree, i.e., have the same product.)
\end{remark}
In order to obtain a category of Bratteli diagrams with premorphisms,
as defined here (this difficulty did not arise  with the definition of \cite{aeg14},
i.e., Definition~\ref{defbd} above),
 we need to divide out by a fairly trivial
 equivalence relation on premorphisms.  (In Definition~\ref{defeq} we will define
a much stronger notion of equivalence which leads to the notion of morphism of
Bratteli diagram.)
\begin{definition}\label{defisopre}
Let $B,C\in \mathbf{BD}_{1}$ and $f,f':B\to C$ be a pair of premorphisms where
$f=(F,(f_{n})_{n=0}^{\infty})$ and
$f'=(F',(f'_{n})_{n=0}^{\infty})$.
Let us say that $f$ is \textit{isomorphic} to $f'$, and write
$f\cong f'$, if $f_{n}=f'_{n}$, $n\geq 0$, and there is a bijective
map from $F$ to $F'$ preserving the range and source maps.
The relation $\cong$ is an equivalence relation on the class of all
premorphisms from $B$ to $C$.
Let us denote the equivalence class of $f$  by
$\overline{f}$.
Let $B$, $C$, and $D$ be objects in $\mathbf{BD}_{1}$ and
let $f:B\to C$ and $g:C\to D$ be premorphisms,
$f=(F,(f_{n})_{n=0}^{\infty})$ and
$g=(G,(g_{n})_{n=0}^{\infty})$
where
$F=\bigcup_{n=0}^{\infty}F_{n}$ and
$G=\bigcup_{n=0}^{\infty}G_{n}$ (disjoint unions).
The composition of $f$ and $g$ is defined as
$gf=(H,(h_{n})_{n=0}^{\infty})$, where $h_{n}=g_{f_{n}}$,
$H=\bigcup_{n=0}^{\infty}H_{n}$, and
$H_{n}=F_{n}\circ G_{f_{n}}$, $n\geq 0$ (i.e., the set of all paths from $s(F_{n})$ to
$r(G_{f_{n}})$; see the remarks before Definition~\ref{defcontr}).
Also, set
$\overline{g}\overline{f}=\overline{gf}$.
\end{definition}

\begin{proposition}\label{propcatpre}
$\mathbf{BD}_{1}$, with  premorphisms modulo the relation (isomorphism)
of Definition~\ref{defisopre}, is a category. (Let us refer to this as the category of Bratteli diagrams
with premorphisms.)
\end{proposition}
\begin{proof}
The proof is essentially the same as that of \cite[Proposition~2.4]{aeg14}.
Note that, although
the relation $(fg)h=f(gh)$ does not strictly speaking hold for premorphisms, we do have
$(\overline{f}\overline{g})\overline{h}=\overline{f}(\overline{g}\overline{h})$.
\end{proof}
\begin{proposition}\label{propisopre}
Two Bratteli diagrams are isomorphic  in the category $\mathbf{BD}_{1}$
with premorphisms (of Proposition~\ref{propcatpre})
if, and only if, they are isomorphic  in the sense of Herman, Putnam, and Skau.
\end{proposition}
\begin{proof}
Suppose that $B=(V,E)$ and $C=(W,S)$ are  Bratteli diagrams which
are  isomorphic  in the category $\mathbf{BD}_{1}$
with premorphisms. Thus there are premorphisms
$f: B\to C$ and $g: C\to B$  such that
$fg\cong\mathrm{id}_{C}$ and $gf\cong\mathrm{id}_{B}$.
Write
$f=(F,(f_{n})_{n=0}^{\infty})$ and
$g=(G,(g_{n})_{n=0}^{\infty})$. We claim that
$f_{n}=g_{n}=n$, for each $n\geq 0$. Let us use induction on
$n$. For $n=0$, by Definition~\ref{defprebd}, $f_0=g_0=0$.
Suppose that $n\geq 1$ and for each $1\leq k\leq n-1$,
$f_{k}=g_{k}=k$. Since $(f_{n})_{n=0}^{\infty}$ and $(g_{n})_{n=0}^{\infty}$ are
increasing  sequences, $f_{n},g_{n}\geq n-1$. If $f_{n}=n-1$ then
$g_{f_{n}}=g_{n-1}=n-1$, which is not the case. Hence $f_{n}\geq n$ and by symmetry
$g_{n}\geq n$. If $f_{n}>n$ then $g_{n}>n$. Then
$n=f_{g_{n}}\geq f_{n}>n$ which is a contradiction. Therefore, $f_{n}=g_{n}=n$
and the claim is proved.

Fix $n\geq 1$. We have $F_{n}\circ G_{n}\cong \mathrm{id}_{W_{n}}$ and
$G_{n}\circ F_{n}\cong \mathrm{id}_{V_{n}}$. Thus
$\mathrm{M}(F_{n})\mathrm{M}(G_{n})={I}_{|W_{n}|}$
and
$\mathrm{M}(G_{n})\mathrm{M}(F_{n})={I}_{|V_{n}|}$
where $|W_{n}|$ denotes the cardinality of $W_{n}$ and similarly for
$V_{n}$. Since the entries of the matrices $\mathrm{M}(G_{n})$ and $\mathrm{M}(F_{n})$
are positive (i.e., non-negative) integers, we have
$|V_{n}|=|W_{n}|$ and the entries of $\mathrm{M}(G_{n})$ and $\mathrm{M}(F_{n})$
are $0$ or $1$ with exactly one $1$ in each row  and each column.
Using this and commutativity of the diagram of $f$ as specified in Definition~\ref{defprebd},
it is not hard to see that
there exist  bijections between $V$ and $W$
and between $E$ and $S$, preserving the gradings and intertwining the
respective source and range maps.

The proof of the backward implication is immediate.
\end{proof}
\begin{definition}\label{defeq}
Let $B, C$ be Bratteli diagrams and $f,g: B\to C$ be premorphisms
with $B=(V,E)$, $C=(W,S)$,
$f=(F,(f_{n})_{n=0}^{\infty})$, and
$g=(G,(g_{n})_{n=0}^{\infty})$.
Let us  say that $f$ is \emph{equivalent} to $g$, and write $f\sim g$,
if there are sequences $(n_{k})_{k=1}^{\infty}$ and $(m_{k})_{k=1}^{\infty}$ of
positive integers such that $n_{k}<m_{k}<n_{k+1}$ and $f_{n_{k}}<g_{m_{k}}<f_{n_{k+1}}$
for each $k\geq 1$, and the diagram

\[
\xymatrix{V_{n_{\resizebox{.28em}{.28em}{1}}}
\ar[r]\ar[d]_{F_{n_{\resizebox{.26em}{.26em}{1}}}}
 & V_{m_{\resizebox{.28em}{.28em}{1}}}
 \ar[r]\ar[d]_{G_{m_{\resizebox{.26em}{.26em}{1}}}}  &
 V_{n_{\resizebox{.28em}{.28em}{2}}}
 \ar[r] \ar[d]_{F_{n_{\resizebox{.26em}{.26em}{2}}}}
 & V_{m_{\resizebox{.28em}{.28em}{2}}}
 \ar[r]\ar[d]_{G_{m_{\resizebox{.26em}{.26em}{2}}}} &\cdots \\
 W_{f_{n_{\resizebox{.26em}{.26em}{1}}}}\ar[r]
 &W_{g_{m_{\resizebox{.26em}{.26em}{1}}}}\ar[r] &
 W_{f_{n_{\resizebox{.26em}{.26em}{2}}}}\ar[r]
 & W_{g_{m_{\resizebox{.26em}{.26em}{2}}}}\ar[r]&\cdots
 }
\]

\noindent commutes, i.e., each minimal square commutes: for each $k\geq 1$,
\begin{equation*}
E_{n_{k},m_{k}}\circ G_{m_{k}}\cong F_{n_{k}}\circ S_{f_{n_{k}},g_{m_{k}}},
\end{equation*}
\begin{equation*}
   E_{m_{k},n_{k+1}}\circ F_{n_{k+1}}\cong
G_{m_{k}}\circ S_{g_{m_{k}},f_{n_{k+1}}}.
\end{equation*}
\end{definition}
At the end of this section we will give two other definitions for equivalence of
premorphisms (Definitions~\ref{defeq2} and \ref{defeq3}).
These may be more natural in some sense (since they do not use subsequences), but we shall
show that all three equivalence relations are the same (Proposition~\ref{proeqpre}).
\begin{lemma}\label{lemeqpre}
Let $B,C\in \mathbf{BD}_{1}$. Then $\sim$ is an equivalence relation on the class of
 premorphisms from $B$ to $C$.
\end{lemma}
\begin{proof}
In view of
the  remark preceding Definition~\ref{defisopre},
the proof is essentially the same as that of \cite[Lemma~2.6]{aeg14}.
\end{proof}
Let us call an equivalence class of premorphisms between Bratteli diagrams $B$ and $C$,
with respect to the relation $\sim$, a \emph{morphism} from $B$ to $C$.
Let us denote the equivalence class of a premorphism
$f:B\to C$  by $[f]:B\to C$, or if there is  no confusion, just   by $f$.

The composition of morphisms $[f]:B\to C$ and $[g]:C\to D$ should be defined as
$[gf]:B\to D$ where $gf$ is the composition of premorphisms. This composition is
well~defined, as is affirmed in  the following theorem.
\begin{theorem}\label{thrbd}
The class $\mathbf{BD}_{1}$, with  morphisms as defined above, is a category.
\end{theorem}
\begin{proof}
The proof is similar to that of \cite[Theorem~2.7]{aeg14} (use
the  remark preceding Definition~\ref{defisopre}).
\end{proof}
Let us refer to $\mathbf{BD}_{1}$, with  morphisms as defined above,
as the  \emph{category of Bratteli diagrams}.
\begin{theorem}\label{thrisomor}
Two Bratteli diagrams are isomorphic  in the category $\mathbf{BD}_{1}$ (Theorem~\ref{thrbd})
if, and only if, they are equivalent  in the sense of Herman, Putnam, and Skau.
\end{theorem}
\begin{proof}
The statement follows from Theorem~\ref{threqubd} and \cite[Theorem~4.5]{aeg14}.
\end{proof}
Here are two  alternative formulations of the definition for equivalence of premorphisms
(Definition~\ref{defeq}).
We shall use the first one in a number of places later.
\begin{definition}\label{defeq2}
Let $f,g: B\to C$ be premorphisms in $\mathbf{BD}_{1}$ such that $B=(V,E)$, $C=(W,S)$,
$f=(F,(f_{n})_{n=0}^{\infty})$, and
$g=(G,(g_{n})_{n=0}^{\infty})$.
Let us say that $f$ is \emph{equivalent} to $g$, in the second sense,
 if for each $n\geq 0$ there is an $m\geq f_{n},g_{n}$ such that
 $F_{n}\circ S_{f_{n},m}\cong G_{n}\circ S_{g_{n},m}$.
\end{definition}
\begin{definition}\label{defeq3}
Let $f,g: B\to C$ be premorphisms in $\mathbf{BD}_{1}$ such that $B=(V,E)$, $C=(W,S)$,
$f=(F,(f_{n})_{n=0}^{\infty})$, and
$g=(G,(g_{n})_{n=0}^{\infty})$.
Let us say that $f$ is \emph{equivalent} to $g$, in the third sense,
 if for each $n\geq 0$ and for each $k\geq n$, there is an
 $m\geq f_{n},g_{k}$ such that $F_{n}\circ S_{f_{n},m}\cong E_{n,k}\circ G_{k}\circ S_{g_{k},m}$.
\end{definition}
Let us show that these two definitions are equivalent to Definition \ref{defeq}.
\begin{proposition}\label{proeqpre}
Definitions \ref{defeq}, \ref{defeq2}, and \ref{defeq3} are
 equivalent.
\end{proposition}
\begin{proof}
The proof is essentially the same as that of \cite[Proposition~2.11]{aeg14}.
\end{proof}
Let us now show that the categories
$\mathbf{BD}	^{1}$ and $\mathbf{BD}_{1}$ are equivalent.
Define the map $\mathcal{F}: \mathbf{BD}^{1}\to \mathbf{BD}_{1}$
as follows. Let $B=(V,E)$ be in $\mathbf{BD}	^{1}$ (see Definition~\ref{defbd} and
the following remarks). Set $\mathcal{F}(B)=(V',E')$ where
$V'=\bigcup_{n=0}^{\infty}V'_{n}$, $E'=\bigcup_{n=1}^{\infty}E'_{n}$,
$V'_{0}=\{(0,1,1)\}$, and for $n\geq 1$, $V'_{n}=\{(n,i,a_{i}) \mid 1\leq i\leq k_{n}\}$ where
$a_{1},a_{2},\ldots,a_{k_{n}}$ are entries of $V_{n}$.
Moreover, $E'_{1}$ consists of all
$(e,f,1),(e,f,2),\ldots,(e,f,m)$ where $e$ is the element of $V'_{0}$,
$f\in V'_{1}$, and $m$ is the number of paths from $e$ to $f$ determined by
$V_{1}$, i.e., $m$ is the third component of $f$.
For $n\geq 2$, $E'_{n}$ consists of all
$(e,f,1),(e,f,2),\ldots,(e,f,m)$ where $e\in V'_{n-1}$, $f\in V'_{n}$, and $m$ is
the number of paths from $e$ to $f$ determined by $E_{n-1}$, i.e., $m$ is
the $ji$th entry of $E_{n-1}$ where $i$ and $j$ are the second components of
$e$ and $f$, respectively.

Let us define the map $\mathcal{F}: \mathbf{BD}^{1}\to \mathbf{BD}_{1}$  on premorphisms  as follows.
Let $f=((F_{n})_{n=1}^{\infty},(f_{n})_{n=1}^{\infty})$ be a premorphism
from $B=(V,E)$ to $C=(W,S)$
in $\mathbf{BD}^{1}$ as in \cite[Definition~2.3]{aeg14}.
Set $\mathcal{F}(f)=((F'_{n})_{n=0}^{\infty},(f_{n})_{n=0}^{\infty})$
where $f_{0}=0$, $F'_{0}=\{(0,0,1)\}$, and for each $n\geq 1$,
$F'_{n}$ consists of all
$(e,e',1),(e,e',2),\ldots,(e,e',m)$ where $e\in V'_{n}$, $e'\in W_{f_n}$, and $m$ is
the number of paths from $e$ to $e'$ determined by $F_{n}$, i.e., $m$ is
the $ji$th entry of $F_{n}$ where $i$ and $j$ are the second components of
$e$ and $e'$, respectively. It is easy to see that
$\mathcal{F}(f)$ is a premorphism in $\mathbf{BD}_{1}$ as in Definition~\ref{defprebd}.
Also, $\mathcal{F}$ maps equivalent premorphisms of $\mathbf{BD}^{1}$
to equivalent  premorphisms of $\mathbf{BD}_{1}$. Hence we can define
$\mathcal{F}$ on morphisms, that is, set
$\mathcal{F}([f])=[\mathcal{F}(f)]$.
\begin{theorem}\label{threqubd}
The map
$\mathcal{F}$ from the category $\mathbf{BD}^{1}$ with morphisms to the category
 $\mathbf{BD}_{1}$ with morphisms, as defined above,
is an equivalence of categories.
\end{theorem}
\begin{proof}
First note that $\mathcal{F}$ is well defined, that is,  $\mathcal{F}$
preserves the identity and composition.
Also it is easily verified that $\mathcal{F}$ is full, faithful, and essentially surjective.
Therefore, by Theorem~\ref{thrcat},
$\mathcal{F}: \mathbf{BD}^{1}\to \mathbf{BD}_{1}$
is an equivalence of categories.
\end{proof}
\begin{remark}
Define the map $\overline{\mathcal{F}}:\mathbf{BD}^{1} \to
\mathbf{BD}_{1}$ from the category $\mathbf{BD}^{1}$ with premorphisms
(see the remarks following Definition~\ref{defbd})
to the category $\mathbf{BD}_{1}$ with equivalence classes of premorphisms
(see Definition~\ref{defisopre} and Proposition~\ref{propcatpre}) as follows. For each $B\in \mathbf{BD}^{1}$
set $\overline{\mathcal{F}}(B)={\mathcal{F}}(B)$ and
 for each premorphism $f$ in $\mathbf{BD}^{1} $ set
$\overline{\mathcal{F}}(f)=\overline{\mathcal{F}(f)}$.
Then $\overline{\mathcal{F}}$ is a functor which is full, faithful, and essentially surjective.
Therefore, it is an equivalence of categories (by Theorem~\ref{thrcat}).
\end{remark}
\section{The Category of Ordered Bratteli Diagrams}
\noindent
In this section we propose a notion of morphism for the class of ordered Bratteli diagrams,
to construct a category of ordered Bratteli diagrams, which we shall denote by $\mathbf{OBD}$.
This construction is very similar to the construction of the category of Bratteli diagrams
$\mathbf{BD}_{1} $  given in the previous section. In particular, first we need the notion of
 premorphism.

 We shall show that  isomorphism in this category coincides with  equivalence of
ordered Bratteli diagrams as defined by Herman, Putnam, and Skau
in \cite{hps92} (see Theorem~\ref{thrisoo} below).
\begin{definition}[\cite{hps92}, Definition~2.3]\label{defobd}
An \textit{ordered Bratteli diagram} is a Bratteli diagram $(V,E)$ together with an order relation
$\geq$ on $E$ such that $e$ and $e'$ are comparable if and only if $r(e)=r(e')$. In other words,
we have a linear order on each set $r^{-1}\{v\}$, $v\in V\setminus V_{0}$.
\end{definition}
The order of an ordered Bratteli diagram induces an order on its contractions.
In fact, if $(V,E,\geq )$ is an ordered Bratteli diagram and $k , l$ are integers with $0\leq k < l$,
 then the set
 $E_{k+1}\circ E_{k}\circ\cdots\circ E_{l}$ of all paths from $V_{k}$ to $V_{l}$
 may be given an induced (lexicographic)
order as follows:
\begin{equation*}
(e_{k+1},e_{k+2},\ldots,e_{l})>(f_{k+1},f_{k+2},\ldots,f_{l})
\end{equation*}
 if for some $i$ with $k+1\leq i\leq l$, $e_{j}= f_{j}$  for $i < j \leq l$
 and $e_{i}>f_{i}$. If $(V,E,\geq )$ is an ordered Bratteli diagram and
$ (V',E')$
 is a
contraction of $(V,E)$ as defined above, then with the induced order $\geq'$,
$(V' ,E' ,\geq' )$  is
again an ordered Bratteli diagram which is called a \textit{contraction} (\textit{telescoping}) of
$(V,E,\geq )$ \cite{dhs99, hps92}.

Herman, Putnam, and Skau consider  a notion of isomorphism of ordered Bratteli diagrams
(similar to their notion of isomorphism of Bratteli diagrams).
Also, they denote by $\approx$  the equivalence relation on ordered Bratteli diagrams generated by
isomorphism and by telescoping \cite{hps92}.
One can see that $B^{1}\approx B^{2}$, where
$B^{1}=(V^{1},E^{1},\geq^{1})$ and
$B^{2}=(V^{2},E^{2},\geq^{2})$, if and only if there exists an ordered Bratteli diagram
$B=(V,E,\geq)$
such that telescoping $B$ to odd levels, $0 < 1 < 3 < \cdots$, yields a telescoping of  $B^{1}$,
and telescoping $B$ to even levels, $0 < 2 < 4 < \cdots$, yields a
telescoping of $B^{2}$ \cite{dhs99}.
This is analogous to the situation for the equivalence relation $\sim$ on Bratteli diagrams as we
discussed in Section~2.

Let us define the category of ordered Bratteli diagrams.
Just as in the construction of the category of Bratteli diagrams in Section~2,
we need a notion of  premorphism before considering the final
notion of morphism (to be called ordered morphism in the sequel) in this category.
Denote by $\mathbf{OBD}$ the class of all ordered Bratteli diagrams.
\begin{definition}[cf.~Definition~\ref{defprebd}]\label{defpreobd}
Let $B=(V,E,\geq)$ and $C=(W,S,\geq)$
be  ordered Bratteli diagrams.
By an (\emph{ordered}) \emph{premorphism} $f: B\to C$ we mean
 a triple $(F, (f_{n})_{n=0}^{\infty},\geq )$ where
$(F, (f_{n})_{n=0}^{\infty})$ is a premorphism according to Definition~\ref{defprebd},
and $\geq$ is a partial order on $F$ such that:

%\vspace{.5em}
(1)
$e,e'\in F$ are comparable if and only if $r(e)=r(e')$, and $\geq$ is a linear order
on $r^{-1}\{v\}$, $v\in W$;

(2)
 the diagram of $f:B\to C$  commutes:

 \[
\xymatrix{V_{0}\ar[r]^{E_{1}}\ar[d]_{F_{0}}
 &V_{1}\ar[r]^-{E_{2}}\ar[d]_{F_{1}} &V_{2}\ar[r]^-{E_{3}}\ar[ld]^{F_{2}} &\cdots \ \ \  \\
 W_{0}\ar[r]_{S_{1}}
 &W_{1}\ar[r]_{S_{2}} &W_{2}\ar[r]_{S_{3}}&\cdots \   .
 }
\]
 Commutativity of the diagram of course amounts to saying that
for each $n\geq 0$,
 $E_{n}\circ F_{n+1}\cong F_{n}\circ S_{f_{n},f_{n+1}}$, which means that
 there is a (necessarily unique) bijective map from $E_{n}\circ F_{n+1}$ to
 $F_{n}\circ S_{f_{n},f_{n+1}}$
 preserving the order and intertwining the respective source and range
maps.
\end{definition}
Similar to Definition~\ref{defisopre}, an isomorphism relation on the class of ordered premorphisms
is defined (considering order).
The proofs of the following propositions are similar to their analogues in Section~2
(Propositions~\ref{propcatpre} and \ref{propisopre}).
\begin{proposition}\label{propcatopre}
The class $\mathbf{OBD}$, with (ordered) premorphisms
modulo the relation (isomorphism)
stated above, is a category. (Let us refer to this as the category of ordered Bratteli diagrams
with premorphisms.)
\end{proposition}
\begin{proposition}\label{propisopr}
Two ordered Bratteli diagrams are isomorphic  in the category $\mathbf{OBD}$ with ordered
premorphisms
if, and only if, they are isomorphic  in the sense of Herman, Putman, and Skau.
\end{proposition}
We may define an equivalence relation on ordered premorphisms in a way similar to Definition~\ref{defeq},
considering order.
Let us call the equivalence classes  \textit{ordered morphisms}
(or if there is no confusion, just \textit{morphisms}) in  $\mathbf{OBD}$.	
Note that  one can also formulate the alternative Definitions~\ref{defeq2} and \ref{defeq3}
for ordered premorphisms (considering order),
and again the analogue of Proposition~\ref{proeqpre} holds, for ordered premorphisms.

The proofs of the following theorems are similar to their analogues in Section~2
(Theorems~\ref{thrbd} and \ref{thrisomor}).
\begin{theorem}\label{thrcatobd}
The class $\mathbf{OBD}$ with (ordered) morphisms as defined above is a category.
\end{theorem}
\begin{theorem}\label{thrisoo}
Two ordered Bratteli diagrams are isomorphic  in the category $\mathbf{OBD}$
if, and only if, they are equivalent  in the sense of Herman, Putnam, and Skau.
\end{theorem}
Let us refer to the category $\mathbf{OBD}$
with  (ordered) morphisms as defined above
as the  \emph{category of ordered Bratteli diagrams}.
%\section{The Category of Essentially Minimal Dynamical Systems and the Functor $\mathcal{P}$}
\section{The Functor $\mathcal{P}$ from $\mathbf{SDS}$ to $\mathbf{OBD}$}
\noindent
In this section, first we define the category of scaled essentially
minimal totally disconnected dynamical systems $\mathbf{SDS}$. Then we construct
a functor $\mathcal{P}$ from $\mathbf{SDS}$ to the category
of ordered Bratteli diagrams $\mathbf{OBD}$.
As we shall show, the essential range of this functor is the full subcategory of $\mathbf{OBD}$
consisting of essentially simple ordered Bratteli diagrams $\mathbf{OBD}_{\mathrm{es}}$.
 We will show that
$\mathcal{P}:\mathbf{SDS}\to \mathbf{OBD}_{\mathrm{es}}$
is an equivalence of categories (Theorem~\ref{threqucat} below).

In this way, we obtain
a functorial formulation and a generalization of one of the main results
of Herman, Putnam, and Skau \cite[Theorem~4.7]{hps92}.
In fact, they  considered only the isomorphism classes of these categories,
but our approach takes into account the morphisms of these categories.

Here we are mainly interested in the minimal systems, but almost all the results hold in
a more general setting, i.e., essentially minimal totally disconnected systems. However, the minimal case will be discussed
specifically at the end of Section~5.
\begin{definition}[cf.~\cite{hps92}, Definition~1.2]\label{defemds}
Let $X$ be a compact  metrizable space. Let
$\varphi$ be a homeomorphism of $X$ and $y\in X$. The triple $(X,\varphi,y)$
is called an \emph{essentially minimal dynamical system} if the dynamical system
$(X,\varphi)$ has a unique minimal (non-empty, closed, invariant) set $Y$ and $y\in Y$.
\end{definition}
\begin{remark}
Note that if   $X$ is totally disconnected and has no isolated points, then $X$ is homeomorphic to
the Cantor set. Moreover, there are essentially minimal  totally disconnected dynamical systems which are not minimal.
For example, the one-point compactification of a locally compact Cantor minimal system
is essentially minimal but not minimal (\cite{ma02})	.
\end{remark}
\begin{definition}\label{defcatds}
Let us define the category of essentially minimal  totally disconnected dynamical systems  $\mathbf{DS}$
as follows. The objects of this category are the essentially minimal  totally disconnected dynamical systems.
Let $(X,\varphi,y)$ and $(Y,\psi,z)$ be in $\mathbf{DS}$. By a morphism
$\alpha :(X,\varphi,y)\to (Y,\psi,z)$ in $\mathbf{DS}$ we mean a homomorphism from
the dynamical system $(X,\varphi)$ to $(Y,\psi)$ (i.e., a continuous map
with $\alpha\circ \varphi=\psi\circ \alpha$) such that $\varphi(y)=z$.
\end{definition}

For the rest of the paper, we shall use the abbreviation t.d.~for totally disconnected.

\begin{remark}
Note that in the above definition, $\alpha$ maps the unique minimal set of $(X,\varphi)$ to
that of $(Y,\psi)$. Also, isomorphism in the category $\mathbf{DS}$ coincides with the notion of
pointed topological conjugacy introduced in \cite{hps92}.
\end{remark}
Let us recall the notion of a Kakutani-Rokhlin partition.
\begin{definition}[\cite{hps92}]\label{defkrp}
Let $(X,\varphi, y)$ be an essentially minimal  t.d. dynamical system.
A \emph{Kakutani-Rokhlin partition} for $(X,\varphi, y)$ is a partition $P$ of $X$ where
\begin{equation*}
P=\{Z(k,j) \mid k=1,\ldots,K,\  j=1,\ldots, J(k)\},
\end{equation*}
in which $K$ and $J(1),\ldots, J(K)$ are non-zero positive  integers and the $Z(k,j)$ are non-empty clopen
subsets of $X$ with the following properties:
\begin{itemize}
\item[(1)]
$\varphi(Z(k,j))=Z(k,j+1)$ for all  $1\leq k\leq K$, and $1\leq j<J(k)$;
\item[(2)] setting $Z=\bigcup_{k}Z(k, J(k))$, we have $y\in Z$ and
$\varphi (Z)=\bigcup_{k}Z(k,1)$.
\end{itemize}
For each $1\leq k\leq K$, the set $\{Z(k,j) \mid   j=1,\ldots, J(k)\}$ is called the $k$th \emph{tower}
of $P$ with \textit{height} $J(k)$. The sets  $Z$ and $\varphi(Z)$  are called
 the \emph{top} and  \emph{base} of $P$, respectively.
\end{definition}
The idea of the following definition was used implicitly in \cite{hps92}.
\begin{definition}\label{defskrp}
Let $(X,\varphi, y)$ be an essentially minimal t.d.~dynamical system.
A \emph{system of Kakutani-Rokhlin partitions} for $(X,\varphi, y)$ is a sequence
$\{{P}_{n}\}_{0}^{\infty}$ of Kakutani-Rokhlin partitions for $X$   such that $P_{0}=\{X\}$ and:
\begin{itemize}
\item[(1)]
if $Z_{n}$ denotes the top of $P_n$ for each $n\geq 1$,
the sequence $\{Z_{n}\}_{n=1}^{\infty}$ is a decreasing
sequence of clopen sets with intersection $\{y\}$;
\item[(2)]
for all $n$, $P_{n}\leq P_{n+1}$, i.e., $P_{n+1}$ is a refinement of $P_n$;
\item[(3)]
$\bigcup_{n=0}^{\infty}P_{n}$ is a base for the topology of $X$.
\end{itemize}
\end{definition}
\begin{remark}
For each essentially minimal t.d.~dynamical system $(X,\varphi, y)$,
  partitions as in Definition~\ref{defskrp} exist in abundance. In fact, let
$\{Z_{n}\}_{n=1}^{\infty}$ be a decreasing
sequence of clopen sets with intersection $\{y\}$. Then, by \cite[Theorem~4.2]{hps92},
there is a (not necessarily unique) system of Kakutani-Rokhlin
partitions for $(X,\varphi, y)$ with $Z_{n}$ the top of the $n$th partition.
\end{remark}
\begin{definition}\label{defSDS}
By a \emph{scaled essentially minimal t.d.~dynamical system} we mean a quadruple
$(X,\varphi,y,\mathcal{R})$ where $(X,\varphi, y)$ is an essentially minimal t.d.
dynamical system and $\mathcal{R}$ is a system of Kakutani-Rokhlin partitions for
$(X,\varphi, y)$. The \textit{category of scaled essentially minimal t.d.~dynamical systems} $\mathbf{SDS}$
is the category whose objects is the class of all essentially minimal t.d.~dynamical systems and
its morphisms are as follows.
Let $(X,\varphi,y,\mathcal{R})$ and $(Y,\psi,z,\mathcal{S})$ be in $\mathbf{SDS}$. By a morphism
$\alpha :(X,\varphi,y,\mathcal{R})\to (Y,\psi,z,\mathcal{S})$ we mean a homomorphism
from the dynamical system $(X,\varphi)$ to $(Y,\psi)$ (i.e., a continuous map
with $\alpha\circ \varphi=\psi\circ \alpha$) such that $\varphi(y)=z$.
\end{definition}
We shall need the following notation in a number of places.
\begin{notation}
Let $(X,\varphi,y)$ be an essentially minimal t.d.~dynamical system
and $P$ and $Q$ be a pair of Kakutani-Rokhlin partitions for it such that
$P\leq Q$ (i.e., $Q$ is a refinement of $P$) and
the top of $P$ contains the top of $Q$.
Considering the elements of $P$ and $Q$ as vertices,
we denote by $E(P,Q)$ the (ordered) edge set from
$P$ to $Q$  defined as follows.
We have an edge in $E(P,Q)$ each time a tower of $Q$ passes
a tower of $P$; specifically, $E(P,Q)$ contains all elements of the form
$(S,T,k)$ where $S=\{Z_{1},\ldots,Z_{n}\}$ and
$T=\{Y_{1},\ldots,Y_{m}\}$ are towers of $P$ and $Q$, respectively,
and $k$ is a non-negative integer such that
$Y_{k+j}\subseteq Z_{j}$ for all $1\leq j\leq n$ (cf.~\cite[Section~4]{hps92}).
Note that $(S,T,k)\in E(P,Q)$ if and only if
$Y_{k+j}\subseteq Z_{j}$ for some $1\leq j\leq n$.
Write $(S,T,k)\leq (S',T',k')$ if $T=T'$ and $k\leq k'$.
This is a partial order on  $E(P,Q)$, which is a total order
on the set of edges leading to a common vertex.
\end{notation}
We shall also need the following lemma.
This is a topological version of \cite[Lemma~3.4]{aeg14}
(see Definition~\ref{defpreobd} for the notation
$\cong$).
\begin{lemma}\label{lemtop}
Let $(X,\varphi,y)$ be an essentially minimal t.d.~dynamical system
and let $P_{1}$, $P_{2}$, and $P_{3}$ be Kakutani-Rokhlin partitions for
it such that $P_{1}\leq P_{2}\leq P_{3}$ and the top of $P_{i}$ contains
the top of $P_{i+1}$, for $i=1,2$.
Then we have $E(P_{1},P_{3})\cong E(P_{1},P_{2})\circ E(P_{2},P_{3})$, i.e.,
the following diagram commutes:

\[
\xymatrix{
P_{1}\ar[rr]^{E(P_{1},P_{2})} \ar[rrd]_{E(P_{1},P_{3})}
&  &P_{2}\ar[d]^{E(P_{2},P_{3})}\\
& & P_{3}\ .
}
\]
\end{lemma}
\begin{proof}
The lemma follows from the following fact.
Let  $P$ and $Q$ be a pair of Kakutani-Rokhlin partitions for $(X,\varphi,y)$ such that
$P\leq Q$  and the top of $P$ contains the top of $Q$.
Let $T=\{Y_{1},\ldots,Y_{n}\}$ be a tower of $Q$.
Then there are unique integers $J_{0}=0< J_{1}<J_{2}<\cdots<J_{m}=n$ and unique
towers $S_{1},S_{2},\ldots,S_{m}$ of $P$ with the following properties.
The height of  $S_{i}$ equals
$h_{i}=J_{i}-J_{i-1}$, $1\leq i\leq m$, and if we set
$S_{i}=\{Z(i,1),Z(i,2)\ldots,Z(i,h_{i})\}$ then
$Y_{J_{i-1}+j}\subseteq Z(i,j)$ for all
 $1\leq j\leq h_{i}$.
Thus, $(S_{i},T,J_{i-1})\in E(P,Q)$ for all $1\leq i\leq m$, and
$(S_{1},T,J_{0})<(S_{2},T,J_{1})<\cdots<(S_{m},T,J_{m-1})$.
\end{proof}

\begin{definition}\label{deffuncop}
Define the contravariant functor $\mathcal{P}: \mathbf{SDS}\to\mathbf{OBD}$
 as follows.
Let $(X,\varphi,y,\mathcal{R})$ be in $\mathbf{SDS}$. Consider
  the ordered
 Bratteli diagram $\mathcal{P}(X,\varphi,y,\mathcal{R})=(V,E,\geq)$
 constructed in \cite[Section~4]{hps92} for $(X,\varphi,y,\mathcal{R})$.
To recall, let $\mathcal{R}$ be as in Definition~\ref{defskrp} and set
$V_{n}=\{(n,T) \mid T \text{ is a tower of } P_{n}\}$,  $n\geq 0$, and
 $V=\bigcup_{n=0}^{\infty}V_{n}$. Set
$E_{n}=\{(n,S,T,k)\mid (S,T,k)\in E(P_{n-1},P_{n})\}$,  $n\geq 1$, and
$E=\bigcup_{n=1}^{\infty}E_{n}$. The partial order on $E$ is defined as the union of partial
orderings on the $E_{n}$ as described
just before Lemma~\ref{lemtop}.

Now let
$(X,\varphi,y,\mathcal{R})$ and  $(Y,\psi,z,\mathcal{S})$ be in
$\mathbf{SDS}$ where $\mathcal{R}=\{P_{n}\}_{n=0}^{\infty}$
and $\mathcal{S}=\{Q_{n}\}_{n=0}^{\infty}$. Let
$\alpha :(X,\varphi,y,\mathcal{R})\to (Y,\psi,z,\mathcal{S})$ be a morphism
in $\mathbf{SDS}$, i.e., a continuous map from $X$ to $Y$ with $\alpha(y)=z$ and
$\alpha\circ \varphi=\psi\circ\alpha$.
Define the ordered premorphism
$f=(F, (f_{n})_{n=0}^{\infty},\geq)$
from
$\mathcal{P}(Y,\psi,z,\mathcal{S})=(W,S,\geq)$ to
$\mathcal{P}(X,\varphi,y,\mathcal{R})=(V,E,\geq)$
as follows.

Set $f_{0}=0$ and $F_{0}=\{0\}$. Suppose that
we have chosen $f_{0},f_{1},\ldots,f_{n-1}$ and $F_{0},F_{1},\ldots,F_{n-1}$
and define $f_{n}$ and $F_{n}$ as follows.
Since $Q_{n}$ is a Kakutani-Rokhlin partition for $(Y,\psi,z)$, the set $\alpha^{-1}(Q_{n})$
of inverse images of elements of $Q_{n}$ is a
Kakutani-Rokhlin partition for $(X,\varphi,y)$.
By the properties~(1) and (3) of Definition~\ref{defskrp},
there is an integer $f_{n}$ with $f_{n}\geq f_{n-1}$ such that
$\alpha^{-1}(Q_{n})\leq P_{f_{n}}$ and the top of
$\alpha^{-1}(Q_{n})$ contains the top of $P_{f_{n}}$.
Set
\[
F_{n}=\{(n,S,T,k)\mid (\alpha^{-1}(S),T,k)\in E(\alpha^{-1}(Q_{n}), P_{f_{n}})\}.
\]
There is a one-to-one correspondence between the elements of
$F_{n}$ and those of
$E(\alpha^{-1}(Q_{n}), P_{f_{n}})$.
Define the order on $F_{n}$ to be the induced order from
$E(\alpha^{-1}(Q_{n}), P_{f_{n}})$.
In this way $F_{n}$ becomes an edge set from $W_{n}$ to $V_{f_{n}}$.

Continuing this procedure, we obtain a cofinal sequence of
integers $(f_{n})_{n=0}^{\infty}$ with $f_{0}=0\leq f_{1}\leq f_{2}\leq\cdots$ and
an  edge set
$F=\bigcup_{n=0}^{\infty} F_{n}$ such that each $F_{n}$ is an
edge set from $W_{n}$ to $V_{f_{n}}$. The source and range   maps
are defined in the natural way, i.e., $s(n,S,T,k)=(n,S)$
and $r(n,S,T,k)=(f_{n},T)$.
The partial order $\leq$ on $F$ is the union of the partial orders on the $F_{n}$.
Now set $f=(F, (f_{n})_{n=0}^{\infty},\geq)$. Applying Lemma~\ref{lemtop},
we see that $f:(W,S)\to (V,E)$ is an (ordered) premorphism.
Set $\mathcal{P}(\alpha)=[f]$ the equivalence class of
$f$.
\end{definition}
 \begin{theorem}
 The map $\mathcal{P}: \mathbf{SDS}\to\mathbf{OBD}$, as defined above,
 is a  contravariant functor.
\end{theorem}
\begin{proof}
First note that $\mathcal{P}$
is single-valued, i.e., with the notation as above,
$\mathcal{P}(\alpha)$ is independent of the choice of
the sequence $(f_{n})_{n=0}^{\infty}$.
To see this, let $(g_{n})_{n=0}^{\infty}$ be another cofinal sequence with
$g_{0}=0\leq g_{1}\leq g_{2}\leq\cdots$ which gives the premorphism
$g=(G,(g_{n})_{n=0}^{\infty})$ according to
Definition~\ref{deffuncop}.
Fix $n\geq 1$ and assume, without loss of generality, that
$f_{n}\leq g_{n}$. Then
$\alpha^{-1}(Q_{n})\leq P_{f_{n}}\leq P_{g_{n}}$. By Lemma~\ref{lemtop},
the  diagram
\[
\xymatrix{
W_{n}\ar[d]_{F_{n}} \ar[rd]^{G_{n}}
&\\
V_{f_{n}}\ar[r]_{E_{f_{n},g_{n}}}
& V_{g_{n}}
}
\]
commutes, which means that
$F_{n}\cong G_{n}\circ E_{f_{n},g_{n}}$.
By the analogous version of Proposition~\ref{proeqpre} for
ordered premorphisms, we have $f\sim g$.

The proof of the fact that $\mathcal{P}$
preserves composition (as a contravariant functor) is similar to
the last part of the proof of \cite[Proposition~3.8]{aeg14},
but much easier (use Lemma~\ref{lemtop}).
\end{proof}
The following theorem is a constructive result. In fact, it is shown that
each premorphism between the Bratteli diagrams of two
essentially minimal t.d.~dynamical systems can be lifted back to a homomorphism
between them. This was shown for the  special case that the
Ver\v{s}ik transformation of the Bratteli diagram of an essentially minimal t.d.~dynamical  system is
isomorphic to that system \cite[Theorem~4.6]{hps92}.
\begin{theorem}\label{thrful}
 The functor $\mathcal{P}: \mathbf{SDS}\to\mathbf{OBD}$
 is a full functor.
\end{theorem}
\begin{proof}
The idea of the proof is to reverse the procedure of Definition~\ref{deffuncop}.
Let $\mathcal{X}_{1}=(X,\varphi,y,\mathcal{R})$ and
$\mathcal{X}_{2}=(Y,\psi,z,\mathcal{S})$
be in $\mathbf{SDS}$ and write
$\mathcal{P}(\mathcal{X}_{1})=(V,E)$ and
$\mathcal{P}(\mathcal{X}_{2})=(W,S)$.
Let $f:(W,S)\to (V,E)$ be an ordered premorphism.
We must show that there is a morphism
$\alpha:\mathcal{X}_{1}\to \mathcal{X}_{2}$ with
$\mathcal{P}(\alpha)=[f]$.

Write $f=(F, (f_{n})_{n=0}^{\infty},\geq)$,
$\mathcal{R}=\{P_{n}\}_{n=0}^{\infty}$, and
$\mathcal{S}=\{Q_{n}\}_{n=0}^{\infty}$.
Let $F=\bigcup _{n=0}^{\infty} F_{n}$ denote the decomposition of $F$ according to
Definition~\ref{defpreobd}.
We have the following fact.
For each $n\geq 0$, $F_{n}$ fills the towers of $P_{f_{n}}$ with the towers of $Q_{n}$;
specifically, let $T$ be a tower of $P_{f_{n}}$. Let $e_{1},e_{2},\ldots, e_{k}$ denote
the edges in $F_{n}$ with range $(f_{n},T)$ and
$e_{1}<e_{2}<\cdots< e_{k}$.
Denote by $S_{i}$  the tower of $Q_{n}$ such that the source of $e_{i}$ is
$(n,S_{i})$, $1\leq i\leq k$. Then the height of $T$ equals the sum of
the heights of  $S_{1},S_{2},\ldots, S_{k}$. (This follows from  the fact
that $f$ is a premorphism by induction on $n$.)

Choose $x$ in $X$. For each $n\geq 0$ there is $A_{n}\in P_{f_{n}}$ such that $x\in A_{n}$.
We have $A_{0}\supseteq A_{1}\supseteq A_{2}\supseteq\cdots$.
Since $X$ is Hausdorff and $\bigcup_{n=0}^{\infty} P_{n}$ is a basis for $X$,
we have $\bigcap_{n=0}^{\infty}A_{n}=\{x\}$.
Fix $n\geq 0$. With $T_{n}$  the tower of $P_{f_{n}}$ containing $A_{n}$,
by the preceding paragraph, there is a unique tower $S_{n}$ in $Q_{n}$ and a unique element $B_{n}$
in $S_{n}$ which corresponds to $A_{n}$ when $F_{n}$ fills $T_{n}$
by the towers of $Q_{n}$. (We will construct $\alpha$ in such a way that
$\alpha(A_{n})\subseteq B_{n}$.)
By Definition~\ref{defpreobd}, for each $n\geq 1$,
we have
$E_{n-1}\circ F_{n}\cong F_{n-1}\circ S_{f_{n-1},f_{n}}$.
Using this, one can see that $B_{n}\subseteq B_{n-1}$.
The set
$\bigcap_{n=0}^{\infty}B_{n}$ is a singleton; let $\alpha(x)$ denote the element of this set.
Thus we have a map $\alpha:X\to Y$.

Our construction yields that $\alpha(A_{n})\subseteq B_{n}$, $n\geq 0$. With this it is not hard to see that
$\alpha$ is continuous. Let us show that $\alpha(y)=z$.
Since $y$ is in the top of each $P_{f_{n}}$,
$\alpha(y)$ is in the top of each $Q_{n}$. Now by the property~(1) in Definition~\ref{defskrp}
we have $\alpha(y)=z$.

Finally let us show that $\alpha\circ \varphi=\psi\circ 	\alpha$. Let $x\in X\setminus \{y\}$.
Choose $A_{n}$'s and $B_{n}$'s as above.
By the property~(1) of Definition~\ref{defskrp}, there is $n_{0}\geq 0$ such that
$A_{n}$  is not contained in the top of $P_{f_{n}}$, for each $n\geq n_{0}$.
Hence $B_{n}$  is not contained in the top of $Q_{n}$, $n\geq n_{0}$.
This shows that $\varphi(A_{n})$ and $\psi(B_{n})$ are elements of
$P_{f_{n}}$ and $Q_{n}$ which are located above
$A_{n}$ and $B_{n}$, respectively, and hence, by the definition of $\alpha$,
$\alpha(\varphi(A_{n}))\subseteq \psi(B_{n})$.
Thus $\alpha(\varphi(x))\in \psi (B_{n})$, $n\geq n_{0}$.
On the other hand, $\psi(\alpha(x))\in \psi(B_{n})$.
Hence,
$\{\alpha(\varphi(x))\}=\bigcap_{n=n_{0}}^{\infty}\psi(B_{n})=\{\psi(\alpha(x))\}$,
i.e., $\alpha(\varphi(x))=\psi(\alpha(x))$.
That $\alpha(\varphi(y))=\psi(\alpha(y))$ follows from the fact that
$\varphi(y)$ is in the base of each $P_{f_{n}}$ and that
$\alpha$ maps the base of $P_{f_{n}}$ into the base of
$Q_{n}$.
Therefore, $\alpha\circ \varphi=\psi\circ 	\alpha$.

This shows that $\alpha:\mathcal{X}_{1}\to \mathcal{X}_{2}$ is a
morphism in $\mathbf{SDS}$ and our construction shows that
$\alpha^{-1}(Q_{n})\leq P_{f_{n}}$, $n\geq 0$.
Moreover, the premorphism associated to $\alpha$ for the sequence
$(f_{n})_{n=0}^{\infty}$ according to Definition~\ref{deffuncop},
is obviously equivalent to $f$. Hence,
$\mathcal{P}(\alpha)=[f]$.
\end{proof}
\begin{lemma}\label{lemfaith}
 The functor $\mathcal{P}: \mathbf{SDS}\to\mathbf{OBD}$
 is faithful.
 \end{lemma}
\begin{proof}
Let $\mathcal{X}_{1}=(X,\varphi,y,\mathcal{R})$ and
$\mathcal{X}_{2}=(Y,\psi,z,\mathcal{S})$
be in $\mathbf{SDS}$ and
$\alpha,\beta: \mathcal{X}_{1}\to \mathcal{X}_{2}$ be  morphisms in
$\mathbf{SDS}$ such that
$\mathcal{P}(\alpha)=\mathcal{P}(\beta) $.
We have to show that $\alpha=\beta$.
Write
$\mathcal{P}(\mathcal{X}_{1})=(V,E)$,
$\mathcal{P}(\mathcal{X}_{2})=(W,S)$,
$\mathcal{R}=\{P_{n}\}_{n=0}^{\infty}$, and
$\mathcal{S}=\{Q_{n}\}_{n=0}^{\infty}$.
Choose a sequence
$(f_{n})_{n=0}^{\infty}$, as in Definition~\ref{deffuncop},
giving rise to an ordered premorphism
$f=(F, (f_{n})_{n=0}^{\infty},\geq)$ for $\alpha$.
Similarly, choose a sequence
$(g_{n})_{n=0}^{\infty}$
giving rise to an ordered premorphism
$g=(G, (g_{n})_{n=0}^{\infty},\geq)$ for $\beta$.
Thus, since $\alpha=\beta$, $f$ is equivalent to $g$.

Choose $x\in X$ and fix $n\geq 0$. There are
$A_{n}\in P_{f_{n}}$ and $A_{n}'\in P_{g_{n}}$
such that $x\in A_{n}$ and $x\in A_{n}'$.
Also, there are $B_{n},B_{n}'\in Q_{n}$ such that
$\alpha(A_{n})\subseteq B_{n}$ and
$\beta(A_{n}')\subseteq B_{n}'$.
By the  version (obvious) of Proposition~\ref{proeqpre} for ordered premorphisms,
there is $m\geq f_{n},g_{n}$ such that
 $F_{n}\circ S_{f_{n},m}\cong G_{n}\circ S_{g_{n}+1,m}$.
 There is $A\in P_{m}$ such that $x\in A$.
 Hence
 $A\subseteq A_{n}$ and  $A\subseteq A_{n}'$.
It is not hard to see that the relation
$F_{n}\circ S_{f_{n},m}\cong G_{n}\circ S_{g_{n}+1,m}$ implies
that $B_{n}=B_{n}'$.
Hence $\{\alpha(x)\}=\bigcap_{n=0}^{\infty} B_{n}=\bigcap_{n=0}^{\infty} B_{n}'
=\{\beta(x)\}$.
Therefore,  $\alpha=\beta$.
\end{proof}
We need to identify the essential range of the functor
$\mathcal{P}: \mathbf{SDS}\to\mathbf{OBD}$
in order to obtain an equivalence of categories theorem.
(By the essential range of a functor we mean those objects in the codomain
category which are isomorphic to the  objects in the range of the functor.)
For an ordered Bratteli diagram
$(V,E,\geq)$, denote by $E_{\max}$ and  $E_{\min}$
the set of maximal and minimal edges of $E$, respectively.
It is easy to see that there is at least one infinitely long path
in $E_{\max}$ and also in  $E_{\min}$.
\begin{definition}[\cite{hps92}, Definition~2.6]
Let  $B=(V,E,\geq)$ be an ordered Bratteli diagram. $B$  is called \textit{essentially simple}
if there are unique infinitely long paths
in $E_{\max}$ and  $E_{\min}$. That is, there is only one sequence
$(e_{1},e_{2},\ldots)$	with each $e_{i}$ in $E_{\max}$
($E_{\min}$, respectively) and $s(e_{i+1})=r(e_{i})$,
for all $i\geq 1$.
\end{definition}
\begin{remark}
Observe that essentially simple ordered Bratteli diagrams correspond to essentially minimal
dynamical systems on compact, totally disconnected metrizable spaces,
simple ordered  Bratteli diagrams (Definition~\ref{defsbd}) correspond to  minimal
dynamical systems on compact, totally disconnected metrizable spaces (Corollary~\ref{cormds}),
and
properly ordered Bratteli diagrams (Definition~\ref{defpobd}) correspond to  minimal
dynamical systems on  the Cantor set (Corollary~\ref{cormdsc}).
\end{remark}
Let us denote by $\mathbf{OBD}_{\mathrm{es}}$
the full subcategory of $\mathbf{OBD}$ containing all essentially simple ordered Bratteli diagrams.
\begin{lemma}\label{lemessran}
The essential range of  $\mathcal{P}: \mathbf{SDS}\to\mathbf{OBD}$
is $\mathbf{OBD}_{\mathrm{es}}$.
\end{lemma}
\begin{proof}
Let $\mathcal{X}$
be in $\mathbf{SDS}$. Then the ordered Bratteli diagram
$\mathcal{P}(\mathcal{X})$
is essentially simple \cite[Section~4]{hps92}.
Now let $B$ be an ordered Bratteli diagram which is isomorphic in $\mathbf{OBD}$ to
$\mathcal{P}(\mathcal{X})$, for some
$\mathcal{X}\in\mathbf{SDS}$.
By Theorem~\ref{thrisoo}, $B$ is equivalent to
$\mathcal{P}(\mathcal{X})$. By \cite[Proposition~2.7]{hps92},
$B$ is also essentially simple. Hence the essential range of
$\mathcal{P}$ is contained in
$\mathbf{OBD}_{\mathrm{es}}$.

Let $B$ be an essentially simple ordered Bratteli diagram.
Denote by $(X,\varphi,y)$  the Ver\v{s}ik transformation associated to
$B$ \cite[Section~3]{hps92}. Fix a system of Kakutani-Rokhlin partitions
$\mathcal{R}$ for
$(X,\varphi,y)$ (which exists by \cite[Theorem~4.2]{hps92}).
Now by \cite[Theorem~4.6]{hps92},
$B$ is equivalent to $\mathcal{P}(X,\varphi,y,\mathcal{R})$.
\end{proof}
Let us recall the following categorical result.
\begin{theorem}[\cite{ma98}, Theorem~IV.4.1]\label{thrcat}
 A functor $F : \mathcal{C} \to \mathcal{D}$ is an equivalence of categories
if and only if it is full, faithful, and essentially surjective, i.e., for each
$d\in \mathcal{D}$ there is a $c\in \mathcal{C}$ such that $d\cong F(c)$.
\end{theorem}
The following theorem is the main result of this section. It is a generalization  of
\cite[Theorem~4.7]{hps92} which states that there is one-to-one correspondence between
equivalence classes of essentially simple ordered Bratteli diagrams and pointed topological conjugacy
classes of essentially minimal t.d.~dynamical systems. In the categorical language,
\cite[Theorem~4.7]{hps92} states that the functor
$\mathcal{P}: \mathbf{SDS}\to\mathbf{OBD}_{\mathrm{es}}$, constructed above,
is a classification functor (see Definition~\ref{defscf}).
However, it is a consequence of Theorem~\ref{threqucat}  that the functor
 $\mathcal{P}: \mathbf{SDS}\to\mathbf{OBD}_{\mathrm{es}}$
is a strong classification functor, since each full and faithful functor is a strong
classification functor \cite[Lemma~5.10]{aeg14}.
\begin{theorem}\label{threqucat}
 The functor $\mathcal{P}: \mathbf{SDS}\to\mathbf{OBD}_{\mathrm{es}}$
 is an equivalence of categories.
\end{theorem}
\begin{proof}
Apply Theorems~\ref{thrful} and \ref{thrcat} and Lemmas~\ref{lemfaith} and
\ref{lemessran}.
\end{proof}
\begin{corollary}
Let $(X,\varphi,y,\mathcal{R})$ and $(Y,\psi,z,\mathcal{S})$ be
in $\mathbf{SDS}$. The following statements are equivalent:
\begin{itemize}
\item[(1)]
$(X,\varphi,y)$ and $(Y,\psi,z)$ are pointed topological conjugate;
\item[(2)]
the ordered Bratteli diagrams
$\mathcal{P}(X,\varphi,y,\mathcal{R})$ and
$\mathcal{P}(Y,\psi,z,\mathcal{S})$  are equivalent in the sense of Herman, Putnam, and Skau;
\item[(3)]
$\mathcal{P}(X,\varphi,y,\mathcal{R})$ is isomorphic to
$\mathcal{P}(Y,\psi,z,\mathcal{S})$ in the category $\mathbf{OBD}$.
\end{itemize}
\end{corollary}
\begin{proof}
This follows from Theorems~\ref{thrisoo} and \ref{threqucat}.
\end{proof}
\section{The Functor $\mathcal{V}$ from $\mathbf{OBD}_{\mathrm{es}}$ to $\mathbf{SDS}$}
\noindent
In this section we shall construct the contravariant functor
$\mathcal{V} : \mathbf{OBD}_{\mathrm{es}}\to \mathbf{SDS}$ which is
the inverse of the functor
$\mathcal{P}: \mathbf{SDS}\to\mathbf{OBD}_{\mathrm{es}}$.
 (Note
that the inverse of an equivalence of categories is unique up to natural isomorphism.)
Our definition
of  $\mathcal{V}$ on objects of $\mathbf{OBD}_{\mathrm{es}}$
coincides with the construction of \cite{hps92}. We define $\mathcal{V}$ on
the morphisms in a  canonical way.

Let $B=(V,E,\geq)$ be an essentially simple ordered Bratteli diagram.
Denote by $\mathcal{V}(B)$ the  Bratteli-Ver\v{s}ik dynamical
system---which is an essentially minimal t.d.~dynamical system---associated
to $B$ as described in \cite[Section~3]{hps92} and  \cite[Section~3]{gps95}.
To recall, $\mathcal{V}(B)$ is defined as follows.
Let $X_{B}$ denote the space of infinite paths, i.e.,
\[
X_{B}=\{(e_{1},e_{2},\ldots) \mid e_{i}\in E_{i}\ \text{and}\ r(e_{i})=s(e_{i+1})\ \text{for all}\
i\geq 1\}.
\]
$X_{B}$ is topologized by specifying a basis of open sets, namely the family of
cylinder sets $U(e_{1},e_{2},\ldots,e_{k})=\{(f_{1},f_{2},\ldots) \mid
f_{i}=e_{i}, \ 1\leq i\leq k\}$.
The resulting topology is  compact and Hausdorff with a countable basis of clopen sets.
 Each cylinder set is  closed in this topology.
The space $X_{B}$ with this topology is called the \textit{Bratteli compactum}
associated to $B$. Denote by $y_{\max}$ and $y_{\min}$ the unique elements of
$E_{\max}$ and $E_{\min}$, respectively. The homeomorphism
$\lambda_{B}:X_{B}\to X_{B}$, called the Ver\v{s}ik transformation,
is constructed as follows. Set $\lambda_{B}(y_{\max})=y_{\min}$.
Let $x=(e_{1},e_{2},\ldots)\neq y_{\max}$ and consider  the smallest
natural number $k$	 with $e_{k}\not\in E_{\max}$.
Let $f_{k}$ denote the successor of $e_{k}$ in $E$ (so that $r(f_{k})=r(e_{k})$).
Let $(f_{1},f_{2},\ldots,f_{k-1})$ denote the unique path in $E_{\min}$ from $v_{0}$,
the unique element of $V_{0}$, to $s(f_{k})$. That is,
$(f_{1},f_{2},\ldots,f_{k})$  is the successor of
$(e_{1},e_{2},\ldots,e_{k})$ in
$E_{1}\circ E_{2}\circ\cdots\circ E_{k}$.
Set
\[
\lambda_{B}(e_{1},e_{2},\ldots)=(f_{1},f_{2},\ldots,f_{k},e_{k+1},e_{k+2},\ldots).
\]
Then $\lambda_{B}$ is a homeomorphism and $(X_{B},\lambda_{B},y_{\max})$
is an essentially minimal t.d.~dynamical system \cite[Section~3]{hps92}.

Let us define a canonical system of Kakutani-Rokhlin partitions
$\mathcal{R}_{B}=\{P_{n}\}_{n=0}^{\infty}$ for $(X_{B},\lambda_{B},y_{\max})$
such that $(X_{B},\lambda_{B},y_{\max}, \mathcal{R}_{B})$ is in $\mathbf{SDS}$.
Set $P_{0}=\{X_{B}\}$. Fix $n\geq 1$ and define $P_{n}$ as follows.
For each $v\in V_{n}$ we have a tower
$T_{v}$ in $P_{n}$.
For each $(e_{1},e_{2},\ldots,e_{n})$ in
$E_{1}\circ E_{2}\circ\cdots\circ E_{n}$ with
$r(e_{n})=v$ we have an element
$U(e_{1},e_{2},\ldots,e_{n})$, as defined above, in $T_{v}$. Hence,
\[
P_{n}=\{U(e_{1},e_{2},\ldots,e_{n}) \mid (e_{1},e_{2},\ldots,e_{n})\in
E_{1}\circ E_{2}\circ\cdots\circ E_{n}\}.
\]
It is not hard to see that each $P_{n}$ is a Kakutani-Rokhlin partition and
that $\mathcal{R}_{B}=\{P_{n}\}_{n=0}^{\infty}$ satisfies the conditions of
Definition~\ref{defskrp} and hence is a system of Kakutani-Rokhlin partitions for
$(X_{B},\lambda_{B},y_{\max})$. Finally, set
$\mathcal{V}(B)=(X_{B},\lambda_{B},y_{\max},\mathcal{R}_{B})$.
Let us summarize these facts.
\begin{proposition}
For each ordered Bratteli diagram $B=(V,E,\geq)$,
$\mathcal{V}(B)=(X_{B},\lambda_{B},y_{\max},\mathcal{R}_{B})$, as defined above,
is in $\mathbf{SDS}$.
\end{proposition}
Let $B=(V,E,\geq)$ be an ordered Bratteli diagram. Define the ordered premorphism
$f_{B}:B\to \mathcal{P}(\mathcal{V}(B))$
as follows. Set
$f_{B}=(F_{B}, (n)_{n=0}^{\infty},\geq)$ where
$F_{B}=\{(v,T_{v})\mid v\in V\}$.
The decomposition of $F_{B}$  is obtained by setting
$F_{B,n}=\{(v,T_{v})\mid v\in V_{n}\}$, $n\geq 0$. The source and range maps of $F_{B}$ are defined by
$s(v,T_{v})=v$ and $r(v,T_{v})=T_{v}$. There is only one way to define an order on $F_{B}$
(since $r^{-1}\{T_{v}\}$ has only one element).
It is not hard to see that
$f_{B}:B\to \mathcal{P}(\mathcal{V}(B))$ is an ordered premorphism
which is an isomorphism in the category of ordered Bratteli diagrams with ordered premorphisms
(cf.~Propositions~\ref{propcatopre} and \ref{propisopr}).
Denote by $\tau_{B}:B\to \mathcal{P}(\mathcal{V}(B))$
the associated ordered morphism, i.e.,
$\tau_{B}=[f_{B}]$, which is an isomorphism of $\mathbf{OBD}$.

Once one fixes the isomorphisms $\tau_{B}=[f_{B}]$, then there is a unique way to
define $\mathcal{V} : \mathbf{OBD}_{\mathrm{es}}\to \mathbf{SDS}$ on morphisms
to be the natural inverse of $\mathcal{P}: \mathbf{SDS}\to\mathbf{OBD}_{\mathrm{es}}$
(see the proof of \cite[Theorem~IV.4.1]{ma98} for details). In fact,
let $h:B\to C$ be a morphism in
$\mathbf{OBD}_{\mathrm{es}}$.
Then $\tau_{C}h\tau_{B}^{-1}: \mathcal{P}(\mathcal{V}(B))\to
\mathcal{P}(\mathcal{V}(C))$ is a morphism in
$\mathbf{OBD}_{\mathrm{es}}$. By Theorem~\ref{thrful} and Lemma~\ref{lemfaith}, there is a
unique morphism $\alpha: \mathcal{V}(C)\to \mathcal{V}(B)$ such that
$\mathcal{P}(\alpha)=h$. Set $\mathcal{V}(h)=\alpha$. Thus, the following diagram commutes:
\[
\xymatrix{
B\ar[r]^(.35){\tau_{B}} \ar[d]_{h} & \mathcal{P}(\mathcal{V}(B))
\ar[d]^{\mathcal{P}(\mathcal{V}(h))}\\
C \ar[r]_(.35){\tau_{C}}& \mathcal{P}(\mathcal{V}(C)).
}
\]
\begin{theorem}\label{thrfuncv}
 The map $\mathcal{V} : \mathbf{OBD}_{\mathrm{es}}\to \mathbf{SDS}$ as defined above,
 is a contravariant functor which is
  an equivalence of categories and the unique (up to natural isomorphism) inverse of
  the functor $\mathcal{P}:\mathbf{SDS}\to \mathbf{OBD}_{\mathrm{es}}$.
\end{theorem}
\begin{proof}
All the required properties of the map $\mathcal{V}$ follow from the equality
$\mathcal{P}(\mathcal{V}(f))\tau_{B}=\tau_{A}h$.
 (See the proof of \cite[Theorem~IV.4.1]{ma98} for details.) In particular, we have
 $\tau: 1_{{\mathbf{OBD}}_{\mathrm{es}}}\cong \mathcal{P}\mathcal{V}$.
\end{proof}
Let us examine how the functor $\mathcal{V}$ acts on morphisms exactly. (This is important when one intends to
apply this functor.) Let $f:B\to C$ be an ordered premorphism as in Definition~\ref{defpreobd}
and set $h=[f]$.
Define a map $\alpha:X_{C}\to X_{B}$ as follows.
Let $x=(s_{1},s_{2},\ldots)$ be in $X_{C}$, i.e., an infinite path in $S$.
Define the path $\alpha(x)=(e_{1},e_{2},\ldots)$ in $X_{B}$ as follows.
Fix $n\geq 0$.
By Definition~\ref{defpreobd}, the following diagram commutes:

 \[
\xymatrix{V_{0}\ar[r]^{E_{0,n}}\ar[d]_{F_{0}}
 &V_{n}\ar[d]^{F_{n}} \\
 W_{0}\ar[r]_{S_{0,f_{n}}}
 &W_{f_{n}}    .
 }
\]

That is, $F_{0}\circ S_{0,f_{n}}\cong E_{0,n}\circ F_{n}$. Thus, there is a unique path
$(e_{1},e_{2},\ldots,e_{n},d_{n})$ in $E_{0,n}\circ F_{n}$  corresponding to
the path $(s_{0},s_{1},\ldots,s_{f_{n}})$ in
$F_{0}\circ S_{0,f_{n}}$,
where $s_{0}$ is the unique element of $F_{0}$.
We need to check that the first $n$ edges of the path associated to each $m>n$ are the same
as those for $n$.
\begin{lemma}\label{lemtech}
With the above notation, let $m>n$ and
 consider the path $(e_{1}',e_{2}',\ldots,e_{m}',d_{m}')$ in
$E_{0,m}\circ F_{m}$
associated
to $m$ according to the above construction.
Then we have $e_{i}'=e_{i}$ for each $1\leq i\leq n$.
\end{lemma}
\begin{proof}
Consider the following commutative diagram:
\[
\xymatrix{
V_{0}\ar[r]^{E_{0,n}}\ar[d]_{F_{0}}
 &V_{n}\ar[d]^{F_{n}}\ar[r]^{E_{n,m}} & V_{m}\ar[d]^{F_{m}} \\
 W_{0}\ar[r]_{S_{0,f_{n}}}
 & W_{f_{n}} \ar[r]_{S_{f_{n},f_{m}}} & W_{f_{m}} .
 }
\]
Since $(e_{1},e_{2},\ldots,e_{n},d_{n})$ in $E_{0,n}\circ F_{n}$
is the unique path corresponding to $(s_{0},s_{1},\ldots,s_{f_{n}})$ in
$F_{0}\circ S_{0,f_{n}}$, we have $r(d_{n})=r(s_{f_{n}})$. Thus the path
\begin{equation}\label{equ1}
(e_{1},e_{2},\ldots,e_{n},d_{n},s_{f_{n}+1},\ldots,s_{f_{m}})
\end{equation}
in $ E_{0,n}\circ F_{n} \circ S_{f_{n},f_{m}}$
is the unique path corresponding to
$(s_{0},s_{1},\ldots,s_{f_{m}})$ in $F_{0}\circ S_{0,f_{m}}$ by the isomorphism
$ E_{0,n}\circ F_{n} \circ S_{f_{n},f_{m}}\cong F_{0}\circ S_{0,f_{m}}$.

On the other hand, the path
$(e_{1}',e_{2}',\ldots,e_{m}',d_{m}')$  in
$E_{0,m}\circ F_{m}$ is the unique path corresponding to
the path $(s_{0},s_{1},\ldots,s_{f_{m}})$
by the isomorphism
$ E_{0,m}\circ F_{m} \cong F_{0}\circ S_{0,f_{m}}$.
Since the  isomorphisms involved are unique (because of the order),
one  sees that in the isomorphism
$ E_{0,m}\circ F_{m} \cong
E_{0,n}\circ F_{n} \circ S_{f_{n},f_{m}}$,
the path $(e_{1}',e_{2}',\ldots,e_{m}',d_{m}')$
corresponds to
the path in (\ref{equ1}). Therefore, $e_{i}'=e_{i}$ for each $1\leq i\leq n$.
\end{proof}
By the preceding lemma, we can define, without ambiguity,
the path $\alpha(x)=(e_{1},e_{2},\ldots)$ in $X_{B}$
associated to the path $x=(s_{1},s_{2},\ldots)$  in $X_{C}$. Thus we have a map
$\alpha:X_{C}\to X_{B}$. It is not hard to see that if we replace $f$ with another representative
of the class $h$, then we get the same $\alpha$.
\begin{proposition}\label{propcom}
Let $h:B\to C$ be an ordered morphism in $\mathbf{OBD}_{\mathrm{es}}$ and $\alpha:X_{C}\to X_{B}$ be its associated map
 as defined above. Then $\mathcal{V}(h)=\alpha$.
\end{proposition}
\begin{proof}
With the notation above, it is easy to see that $\alpha(U(s_{1},s_{2},\ldots,s_{f_{n}}))
\subseteq U(e_{1},e_{2},\ldots,e_{n})$. This shows that $\alpha$ is continuous.
Moreover, we have $\alpha\circ \lambda_{C}=\lambda_{B}\circ \alpha$
and $\alpha$ maps the unique path in $S_{\max}$ to the unique path in $E_{\max}$.
Thus $\alpha:\mathcal{V}(C)\to \mathcal{V}(B)$ is a morphism in $\mathbf{SDS}$.
On the other hand, it is easily verified that
$\mathcal{P}(\alpha)\tau_{B}=\tau_{A}h$. Hence
$\mathcal{P}(\alpha)=\mathcal{P}(\mathcal{V}(h))$ and by Lemma~\ref{lemfaith},
we get $\mathcal{V}(h)=\alpha$.
\end{proof}
As stated in the proof of Theorem~\ref{thrfuncv}, the correspondence $\tau$, as defined above,
gives $1_{\mathbf{OBD}_{\mathrm{es}}}\cong \mathcal{P}\mathcal{V}$.
Using this, a standard categorical procedure gives a correspondence $\sigma$
which implements $1_{\mathbf{SDS}}\cong \mathcal{V}\mathcal{P}$.
In fact, let $\mathcal{X}$ be in $\textbf{SDS}$. Thus,
$\tau_{\mathcal{P}(\mathcal{X})}: \mathcal{P}(\mathcal{X})\to
\mathcal{P}(\mathcal{V}(\mathcal{P}(\mathcal{X})))$ is an isomorphism in $\mathbf{OBD}$.
By Theorem~\ref{thrful} and Lemma~\ref{lemfaith}, there is a unique isomorphism
$\sigma_{\mathcal{X}}:\mathcal{X}\to \mathcal{V}(\mathcal{P}(\mathcal{X}))$
such that
$\mathcal{P}(\sigma_{\mathcal{X}})=\tau_{\mathcal{P}(\mathcal{X})}^{-1}$.
Moreover, we have
$\sigma : 1_{\mathbf{SDS}}\cong \mathcal{V}\mathcal{P}$.
Let us  describe how $\sigma$ acts exactly.

Let $\mathcal{X}=(X,\varphi,y,\mathcal{R})$ be in $\mathbf{SDS}$ where
$\mathcal{R}=\{P_{n}\}_{n=0}^{\infty}$. Choose $x\in X$.
For each $n\geq 0$, let $S_{n}$ denote the tower of $P_{n}$ containing $x$.
Fix $n\geq 1$ and denote by $k_{n}$  the height at which $S_{n}$ passes through
$S_{n-1}$ (see Definition~\ref{deffuncop}). Set $e_{n}=(n,S_{n-1},S_{n},k_{n})$ and
$\alpha(x)=(e_{1},e_{2},\ldots)$.
Thus we have a map $\alpha:X\to X_{B}$ where $B=(V,E)=\mathcal{P}(\mathcal{X})$.
\begin{lemma}
With the above notation, we have
$\sigma_{\mathcal{X}}=\alpha$.
\end{lemma}
\begin{proof}
First we show that
$\alpha: \mathcal{X}\to \mathcal{V}(\mathcal{P}(\mathcal{X}))$
is a morphism in $\mathbf{SDS}$.
To this end, we need to show that $\alpha$ is continuous, $\alpha(y)=y_{\max}$, and
$\alpha\circ\varphi=\lambda_{B}\circ\alpha$.
Let $x\in X$ and $\alpha(x)=(e_{1},e_{2},\ldots)$ be as above.
For each $n\geq 1$, let $A_{n}$ denote the element
of the tower $S_{n}$ such that $x\in A_{n}$. Then we have $A_{1}\supseteq A_{2}\supseteq\cdots$.
It follows that $\alpha(A_{n})\subseteq U(e_{1},e_{2},\ldots,e_{n})$, $n\geq 1$.
This proves that $\alpha$ is continuous. The equalities $\alpha(y)=y_{\max}$ and
$\alpha\circ\varphi(y)=\lambda_{B}\circ\alpha(y)$ are easy to check.
Let $x\in X\setminus\{y\}$ and $k$ be the smallest natural number such that
$x$ is not in the top of $P_{k}$. Let $\alpha(\varphi(x))=(f_{1},f_{2},\ldots)$.
Then, for each $n\geq k$, both $x$ and $\varphi(x)$ are in the same tower of $P_{n}$ (i.e., the tower $S_{n}$).
Hence $f_{n}=e_{n}$, for each $n\geq k+1$. On the other hand, for each $1\leq n\leq k-1$,
since $x$ is in the top of $P_{n}$, $\varphi(x)$ is in the bottom of $P_{n}$.
This shows that $(f_{1}, f_{2},\ldots,f_{k})$ is
is the successor of
$(e_{1},e_{2},\ldots,e_{k})$ in
$E_{1}\circ E_{2}\circ\cdots\circ E_{k}$.
Therefore,
$\alpha(\varphi(x))=\lambda_{B}(\alpha(x))$.

To see that $\sigma_{\mathcal{X}}=\alpha$ it is enough to show that
$\tau_{\mathcal{P}(\mathcal{X})}\mathcal{P}(\alpha)=\mathrm{id}_{C}$ where
$C=\mathcal{P}(\mathcal{V}(\mathcal{P}(\mathcal{X})))$ (since $\mathcal{P}$ is faithful).
Observe that $\alpha$ is bijective and with the above notation,
$\alpha(A_{n})= U(e_{1},e_{2},\ldots,e_{n})$, $n\geq 1$.
Let $\mathcal{R}_{B}=\{Q_{n}\}_{n=0}^{\infty}$ denote the scale of the system
$\mathcal{V}(\mathcal{P}(\mathcal{X}))$. Then we have $\alpha(P_{n})=Q_{n}$, $n\geq 0$.
Thus  in applying Definition~\ref{deffuncop}
to obtain  $\mathcal{P}(\alpha)$
we can take $f_{n}=n$, $n\geq 0$.
Using this it is not hard to see that $\tau_{\mathcal{P}(\mathcal{X})}\mathcal{P}(\alpha)=\mathrm{id}_{C}$.
\end{proof}
Let us summarize the results of Sections~4 and 5.
\begin{theorem}\label{theequcat2}
The contravariant functors
$\mathcal{P}: \mathbf{SDS}\to\mathbf{OBD}_{\mathrm{es}}$ and
$\mathcal{V} : \mathbf{OBD}_{\mathrm{es}}\to \mathbf{SDS}$
are equivalence of categories which are inverse of each other and
$\tau: 1_{\mathbf{OBD}_{\mathrm{es}}}\cong \mathcal{P}\mathcal{V}$ and
$\sigma : 1_{\mathbf{SDS}}\cong \mathcal{V}\mathcal{P}$.
\end{theorem}
\section{Cantor Minimal Systems}
\noindent
In this section we apply the results of Sections~4 and 5 to Cantor minimal dynamical systems.
First, we investigate the minimal dynamical systems on compact, totally disconnected metrizable spaces
(Corollary~\ref{cormds}). Then we will deal with minimal dynamical systems on  Cantor sets (i.e.,
compact, totally disconnected metrizable spaces with no isolated points---which we know are all
 homeomorphic---if they are non-empty!).

\begin{definition}[\cite{dhs99}, Section~2.1]\label{defsbd}
Let  $B=(V,E)$ be a  Bratteli diagram (as in Definition~\ref{defbd2}).
$B$ is called \textit{simple} if there exists a telescoping
$(V',E')$ of $(V,E)$ such that the embedding matrices of $(V',E')$
have only non-zero elements in each level. In other words, $B$ is simple if for each
$n\geq 0$ there is $m>n$ such that for every   $v\in V_{n}$ and for every
$w\in V_{m}$ there is a path in $E_{n,m}$ from $v$ to $w$.
\end{definition}
The following is part of the literature  (it follows from the results of \cite{hps92}).
\begin{proposition}
Let $B=(V,E,\geq)$ be an essentially simple ordered Bratteli diagram. Then the following
statements are equivalent:
\begin{itemize}
\item[(1)]
the system
$(X_{B},\lambda_{B})$ is minimal;
\item[(2)]
 $(V,E)$ is a simple Bratteli diagram.
\end{itemize}
\end{proposition}
Denote by
$\mathbf{OBD}_{\mathrm{ess}}$ the full subcategory of
$\mathbf{OBD}_{\mathrm{es}}$ consisting of essentially simple ordered Bratteli diagrams
which are simple. Also, denote by $\mathbf{SDS}_{\mathrm{m}}$
the full subcategory of
$\mathbf{SDS}$ consisting of  scaled minimal dynamical systems on
compact, totally disconnected metrizable spaces. The following statement follows from
Theorem~\ref{theequcat2}.
\begin{corollary}\label{cormds}
The categories  $\mathbf{SDS}_{\mathrm{m}}$ and $\mathbf{OBD}_{\mathrm{ess}}$,
as defined above, are equivalent. More precisely,
the contravariant functors
$\mathcal{P}: \mathbf{SDS}_{\mathrm{m}}\to\mathbf{OBD}_{\mathrm{ess}}$ and
$\mathcal{V} : \mathbf{OBD}_{\mathrm{ess}}\to \mathbf{SDS}_{\mathrm{m}}$
are  inverse of each other.
\end{corollary}
Recall that for a Bratteli diagram $B=(V,E)$, the Bratteli compactum $X_{B}$ (defined at the beginning of
Section~5) is a compact, totally disconnected metrizable space.
Thus, to obtain a Cantor set we need only consider the property
oh having no isolated points and its meaning on the diagram $B$.
\begin{lemma}\label{lembcan}
Let $B=(V,E)$ be a Bratteli diagram.
Then the following
statements are equivalent:
\begin{itemize}
\item[(1)]
$X_{B}$ is homeomorphic to the Cantor set;
\item[(2)]
for each infinite path $x=(e_{1},e_{2},\ldots)$ in $X_{B}$ and each $n\geq 1$
there is an infinite path $y=(f_{1},f_{2},\ldots)$ with
$x\neq y$ and $e_{k}=f_{k}$, $1\leq k\leq n$;
\item[(3)]
for each
$n\geq 0$ and each  $v\in V_{n}$ there is $m\geq n$ and $w\in V_{m}$
such that there is path from $v$ to $w$ and $|s^{-1}(\{w\})|\geq 2$.
\end{itemize}
\end{lemma}
\begin{proof}
In view of the definition of the topology of $X_{B}$ (Section~5),
(2) is equivalent to having no isolated points. Thus (1) is equivalent to (2). Also, it is easy to see that
(3) is just a reformulation of (2).
\end{proof}
\begin{proposition}\label{probcan}
Let $B=(V,E)$ be a simple Bratteli diagram. Then the following
statements are equivalent:
\begin{itemize}
\item[(1)]
$X_{B}$ is homeomorphic to the Cantor set;
\item[(2)]
$X_{B}$ is infinite;
\item[(3)]
the set $\{n\in\mathbb{N} \mid |E_{n}|\geq 2\}$ is infinite.
\end{itemize}
\end{proposition}
\begin{proof}
It is immediate that (1) implies (2) and also (2) implies (3). Moreover,
it is not hard to see that (3) and the fact that $B$ is simple imply item~(3) of
Lemma~\ref{lembcan}.
\end{proof}
\begin{definition}[\cite{dhs99}]\label{defpobd}
Let $B=(V,E,\geq)$ be an essentially simple ordered Bratteli diagram. $B$  is called \emph{properly ordered} if it is
simple and $X_{B}$ is infinite (cf.~Proposition~\ref{probcan}).
\end{definition}
Denote by
$\mathbf{OBD}_{\mathrm{po}}$ the full subcategory of
$\mathbf{OBD}_{\mathrm{es}}$ consisting of properly ordered Bratteli diagrams.
Also, denote by $\mathbf{SDS}_{\mathrm{cm}}$
the full subcategory of
$\mathbf{SDS}$ consisting of  scaled minimal dynamical systems on
Cantor sets. The following statement follows from
Theorem~\ref{theequcat2}.
\begin{corollary}\label{cormdsc}
The categories  $\mathbf{SDS}_{\mathrm{cm}}$ and $\mathbf{OBD}_{\mathrm{po}}$,
as defined above, are equivalent. More precisely,
the contravariant functors
$\mathcal{P}: \mathbf{SDS}_{\mathrm{cm}}\to\mathbf{OBD}_{\mathrm{po}}$ and
$\mathcal{V} : \mathbf{OBD}_{\mathrm{po}}\to \mathbf{SDS}_{\mathrm{cm}}$
are  inverse of each other.
\end{corollary}
%\section{Factors an Extensions}
\section{The Direct Limit Functor}
\noindent
 In this section we define the functor
 $\mathcal{D}:\mathbf{BD}\to \mathbf{DG}$ from the category of Bratteli diagrams to the category
 of scaled dimension groups. It is well known that one can associate to each
 Bratteli diagram $B$ a dimension group, say $\mathcal{D}(B)$ \cite{el76, hps92}.
The nontrivial part is  how to define the map $\mathcal{D}$ on
morphisms of $\mathbf{BD}$ in such a way that $\mathcal{D}:\mathbf{BD}\to \mathbf{DG}$
is  a functor, and in fact an equivalence of categories. Although it
is already known that the category of Bratteli diagrams and
the category of scaled dimension groups are equivalent
(\cite[Corollary~6.5]{aeg14}), we give an explicit functor giving this equivalence.
 (The idea of the construction of this
functor was mentioned in the proof of \cite[Corollary~6.5]{aeg14}.)

Let us recall some definitions about dimension groups. By an \emph{ordered group}
we mean a pair $(G,G^{+})$ where $G$ is a  abelian group and $G^{+}$
is a \emph{(positive) cone} for $G$, i.e., $G^{+}+G^{+}\subseteq G^{+}$,
$G^{+}-G^{+}= G$, and $G^{+}\cap (-G^{+})=\{0\}$.
The partial order induced by $G^{+}$ on $G$ is defined by
$x\geq y$ if $x-y\in G^{+}$. By an \emph{order unit} for $(G,G^{+})$
we mean an element $u$ in $G^{+}$ such that, for every g in $G^{+}$,
$nu\geq g$ for some $n$ in $\N$. For each $m\in\N$, the standard cone of the group
$\Z^{m}$ is defined by
$(\Z^{m})^{+}=\{(n_{1},\ldots,n_{m})\mid n_{i}\geq 0, \ 1\leq i\leq m\}$.

A \emph{dimension group} is a countable ordered group obtained as a direct limit
of a sequence of finitely generated free abelian groups with standard order and positive
group homomorphisms as maps \cite{el76}.
It is known that a countable ordered group $(G,G^{+})$ is a dimension group if, and only if,
it is unperforated (i.e., if $na\geq 0$ for some $n\in\N$, then $a\geq 0$, $a\in G$)
and has the Riesz interpolation property, i.e., if $a_{1},a_{2},b_{1},b_{2}\in G$
with $a_{i}\leq b_{j}$, $1\leq i,j\leq 2$, then there exists $c\in G$ with $a_{i}\leq c\leq b_{j}$,
$1\leq i,j\leq 2$ \cite{ehs80}.

A \emph{scale} for an ordered group $(G,G^{+})$ is a subset $\Gamma$ of
$G^{+}$ which is generating, hereditary, and directed \cite{ef81}.
Let us denote by
$\mathbf{DG}$
the category of  scaled dimension groups (i.e., triples of the form
$(G,G^{+}, \Gamma)$ where $(G,G^{+})$ is a dimension group and
$\Gamma$ is a scale) with order and scale preserving  homomorphisms
(see \cite{el76, ehs80, ef81, wo93}).
\begin{definition}\label{defdobj}
Let $B=(V,E)$ be  in $\mathbf{BD}$ (i.e., a Bratteli diagram as in Definition~\ref{defbd})
and let us define a scaled dimension group $\mathcal{D}(B)=(G,G^{+}, \Gamma)$ as follows.
Write $V=(V_{n})_{n=1}^{\infty}$, $E=(E_{n})_{n=1}^{\infty}$, and
$V_{n}^{T}=(m_{n1}\ m_{n2}\ \cdots\ m_{nk_{n}})$, $n\geq 1$.
Set $G_{n}=\mathbb{Z}^{k_{n}}$ and let $G_{n}^{+}$ denote its standard cone. Set
\[
\Gamma_{n}=\{(a_{1},a_{2},\ldots,a_{k_{n}})\in G_{n}\mid  0\leq a_{i}\leq m_{ni},\ 1\leq i\leq k_{n} \}.
\]
Thus, $(G_{n},G_{n}^{+}, \Gamma_{n})$ is a scaled dimension group. Let
$\varphi_{n}:G_{n}\to G_{n+1}$ be the homomorphism defined by the multiplication by $E_{n}$.
Observe that $\varphi_{n}(\Gamma_{n})\subseteq \Gamma_{n+1}$, since $E_{n}V_{n}\leq V_{n+1}$.
Set $\mathcal{D}(B)=\varinjlim (G_{n},\varphi_{n})$. More precisely, $\mathcal{D}(B)$
is defined as follows.  Set
\[
G_{\infty}=\left\{(a_{n})\in \prod_{n=1}^{\infty}G_{n}\mid \exists\, m\geq 1\ \
a_{n+1}=\varphi_{n}(a_{n}) \ (n\geq m)\right\},
\]
and let $G_{0}$ be the subgroup of $G_{\infty}$ consisting of sequences which are zero everywhere except in
finitely many entries. Set $G=G_{\infty}/G_{0}$. Moreover, let $\pi:G_{\infty}\to G$ be the quotient map and
set $G^{+}=\pi(G_{\infty}\cap \prod_{n=1}^{\infty} G_{n}^{+})$ and
$\Gamma=\pi(G_{\infty}\cap \prod_{n=1}^{\infty} \Gamma_{n})$.
Set $\mathcal{D}(B)=(G,G^{+}, \Gamma)$ which is a scaled dimension group
and so is in $\mathbf{DG}$.
%\[\varphi_{n}(a_{1},\ldots,a_{k_{n}})=E_{n}\left(
% \begin{smallmatrix}
% a_{1}\\
% \vdots\\
% a_{k_{n}}
 %\end{smallmatrix}
% \right).
% \]
\end{definition}
\begin{definition}\label{defdmor}
Let $B,C\in \mathbf{BD}$ and
$h:B\to C$ be a morphism in $\mathbf{BD}$ (see \cite[Section~2]{aeg14} for definition).
Let us define a morphism
$\mathcal{D}(h):\mathcal{D}(B)\to \mathcal{D}(C)$ in  $\mathbf{DG}$  as follows.
Let $f$ be a premorphism in   $\mathbf{BD}$ such that $h=[f]$.
Write $B=(V,E)$, $C=(W,S)$, and $f=\left((F_{n})_{n=1}^{\infty},(f_{n})_{n=1}^{\infty}\right)$.
Let
$\mathcal{D}(B)=(G,G^{+}, \Gamma)$ and $\mathcal{D}(C)=(H,H^{+}, \Lambda)$
be as in Definition~\ref{defdobj}, with connecting maps $(\varphi_{n})_{n=1}^{\infty}$ and
$(\psi_{n})_{n=1}^{\infty}$, respectively.
Let
$\eta_{n}:G_{n}\to H_{f_{n}}$ denote the homomorphism defined by the multiplication by  $F_{n}$.
Thus we have the following commutative diagram:

 \[
\xymatrix{G_{1}\ar[r]^{\varphi_{1}}\ar[d]_{\eta_{{1}}}
 &G_{2}\ar[r]^-{\varphi_{2}}\ar[d]_{\eta_{{2}}} &G_{3}\ar[r]^-{\varphi_{3}}\ar[ld]^{\eta_{{3}}}
 &\cdots \ \ \  \\
 H_{1}\ar[r]_{\psi_{1}}
 &H_{2}\ar[r]_{\psi_{2}} &H_{3}\ar[r]_{\psi_{3}}&\cdots \   .
 }
\]

\noindent
Therefore, we obtain a group homomorphism $\eta:G\to H$ with
$\eta(G^{+})\subseteq H^{+}$ and $\eta(\Gamma)\subseteq \Lambda$.
More precisely, for each $(a_{n})_{n=1}^{\infty}$ in $G_{\infty}$ let
$\eta ((a_{n})_{n=1}^{\infty}+G_{0})=((b_{n})_{n=1}^{\infty}+H_{0})$
be defined as follows. Let $m$ be a natural number such that $a_{n+1}=\varphi_{n}(a_{n})$,
$n\geq m$. Set $b_{1}=b_{2}=\cdots=b_{f_{m}-1}=0$,
$b_{f_{n}}=\eta_{n}(a_{n})$ for $n\geq m$, and
$b_{k}=\varphi_{k-1}\varphi_{k-2}\cdots\varphi_{f_{n}}(b_{f_{n}})$ for $n\geq m$ and
$f_{n}<k<f_{n+1}$. It is not hard to see that $\eta$ has the stated properties.
Note that $\eta$ does not depend on the representative $f$ of the equivalence class $h$
(see the proof of Proposition~\ref{profund} below).
Now set $\mathcal{D}(h)=\eta$ which is a morphism from
$\mathcal{D}(B)$ to $\mathcal{D}(C)$ in the category $\mathbf{DG}$.
\end{definition}


\begin{proposition}\label{profund}
The map $\mathcal{D}:\mathbf{BD}\to \mathbf{DG}$, as defined above, is a functor.
\end{proposition}
\begin{proof}
First note that, with the notation of Definition~\ref{defdmor},
$\mathcal{D}(h)$ does not depend on the representative $f$ of the equivalence class $h$.
This fact  follows easily from \cite[Proposition~2.11]{aeg14}.
That $\mathcal{D}$ preserves composition and identity is also easy to check.
Therefore, $\mathcal{D}$ is a functor.
\end{proof}

\begin{theorem}\label{thmd}
The functor $\mathcal{D}:\mathbf{BD}\to \mathbf{DG}$
is an equivalence of categories.
\end{theorem}

\begin{proof}
We will show that the functor $\mathcal{D}$ is full, faithful, and essentially surjective.
Then the statement  follows from Theorem~\ref{thrcat}. First let us show that
$\mathcal{D}$ is faithful. Let $B,C\in \mathbf{BD}$ and
$h,k:B\to C$ be morphisms in $\mathbf{BD}$ such that
$\mathcal{D}(h)=\mathcal{D}(k)$. Let
$f=\left((F_{n})_{n=1}^{\infty},(f_{n})_{n=1}^{\infty}\right)$ and
$g=\left((K_{n})_{n=1}^{\infty},(g_{n})_{n=1}^{\infty}\right)$ be premorphisms
such that $h=[f]$ and $k=[g]$. We have o show that
$f$ is equivalent to $g$.

Write $B=(V,E)$ and $C=(W,S)$. Let
$\mathcal{D}(B)=(G,G^{+}, \Gamma)$ and $\mathcal{D}(C)=(H,H^{+}, \Lambda)$
as in Definition~\ref{defdobj}, with connecting maps $(\varphi_{n})_{n=1}^{\infty}$ and
$(\psi_{n})_{n=1}^{\infty}$, respectively.
Fix $n\geq 1$ and let $m\geq 1$ be such that $G_{n}=\mathbb{Z}^{m}$.
Moreover, let $\{e_{1},e_{2},\ldots,e_{m}\}$ denote the standard basis of $G_{n}$ (as a $\mathbb{Z}$-module).
Define $x_{1},x_{2},\ldots,x_{m}\in G_{\infty}$ as follows. Write
$x_{i}=(x_{ik})_{k=1}^{\infty}$ such that
$x_{i1}=x_{i2}=\cdots=x_{i(n-1)}=0$, $x_{in}=e_{i}$, and
$x_{ik}=\varphi_{n,k}(e_{i})$ for $k>n$, where
$\varphi_{n,k}=\varphi_{k-1}\varphi_{k-2}\cdots \varphi_{n}$ which is a homomorphism
from $G_{n}$ to $G_{k}$. Let $\eta_{n}:G_{n}\to H_{f_{n}}$ and
$\theta_{n}:G_{n}\to H_{g_{n}}$ be the maps obtained by applying Definition~\ref{defdmor}
to $f$ and $g$, respectively. Since
$\mathcal{D}(h)(x_{i}+G_{0})=\mathcal{D}(k)(x_{i}+G_{0})$, $1\leq i\leq m$, there is
$l\geq f_{n},g_{n}$ such that
$\psi_{f_{n},l}\circ\eta_{n}(e_{i})=\psi_{g_{n},l}\circ\theta_{n}(e_{i})$, $1\leq i\leq m$.
Thus, $\psi_{f_{n},l}\circ\eta_{n}=\psi_{g_{n},l}\circ\theta_{n}$
and hence
$S_{f_{n}l}F_{n}=S_{g_{n}l}K_{n}$. By \cite[Proposition~2.11]{aeg14},
$f$ is equivalent to $g$. Therefore, $\mathcal{D}$ is faithful.

Next let us show that $\mathcal{D}$ is full. In fact, the proof is based on a one-sided
intertwining argument. Let $B,C\in \mathbf{BD}$ and
$\varphi:\mathcal{D}(B)\to \mathcal{D}(C)$ be a morphism in $\mathbf{DG}$.
We have to show that there is a morphism $h:B\to C$ in $\mathbf{BD}$ such that
$\mathcal{D}(h)=\varphi$. We retain the notation of the preceding paragraph.
Let   $\varphi^{n}:G_{n}\to G$ and $\psi^{n}:H_{n}\to H$, $n\geq 1$, be
the homomorphisms coming from the construction of the direct limit in Definition~\ref{defdobj}.
Then we have
$\Gamma=\bigcup_{n=1}^{\infty}\varphi^{n}(\Gamma_{n})$ and
$\Lambda=\bigcup_{n=1}^{\infty}\psi^{n}(\Lambda_{n})$. Since each $G_{n}$ is finitely generated,
there is a strictly increasing sequence $(f_{n})_{n=1}^{\infty}$ in $\mathbb{N}$
such that $\varphi (\varphi^{n}(\Gamma_{n}))\subseteq \psi^{f_{n}}(\Lambda_{f_{n}})$, $n\geq1$.
Thus for each $n\geq1$,  there is a (not necessarily unique) order and scale preserving homomorphism
$\theta_{n}:G_{n}\to H_{f_{n}}$ such that
$\varphi\varphi^{n}=\psi^{f_{n}}\theta_{n}$.
Consider the following (possibly non-commutative)
 diagram:
\[
\xymatrix{
{G_{n}}\ar[d]_{\theta_{n}}\ar[r]^{\varphi_{n}} & G_{n+1}
\ar[r]^{\varphi^{n+1}}\ar[d]^{\theta_{n+1}}
 & G
  \ar[d]^{\varphi}\\
H_{f_{n}}\ar[r]_{\psi_{f_{n},f_{n+1}}} &H_{f_{n+1}}\ar[r]_{\psi^{f_{n+1}}}& H\, .
}
\]
The right hand side square commutes. Also, if we omit $\theta_{n+1}$ the remaining diagram
commutes. Thus, $\psi^{f_{n+1}}\theta_{{n+1}}\varphi_{n}=\psi^{f_{n+1}}\psi_{f_{n},f_{n+1}}\theta_{n}$.
Again since $G_{n}$ is finitely generated, there is $m_{n}\geq f_{n+1}$ such that
$\psi_{f_{n+1},m_{n}}\theta_{{n+1}}\varphi_{n}=\psi_{f_{n+1},m_{n}}\psi_{f_{n},f_{n+1}}\theta_{n}
=\psi_{f_{n},m_{n}}\theta_{n}$ as maps from $G_{n}$ to $H_{m_{n}}$. This shows that
we can (so let us) replace the sequence $(f_{n})_{n=1}^{\infty}$ with another sequence
(denoted with the same notation) and the maps
$\theta_{n}$'s with $\eta_{n}$'s such that the left hand side square in the above diagram commutes
(for each $n\geq 1$). Thus we obtain a strictly increasing sequence $(f_{n})_{n=1}^{\infty}$
and order and scale preserving homomorphisms $\eta_{n}:G_{n}\to H_{f_{n}}$, $n\geq 1$,
such that $\varphi\varphi^{n}=\psi^{f_{n}}\eta_{n}$ and
$\eta_{n+1}\varphi_{n}=\psi_{f_{n},f_{n+1}}\eta_{n}$.
For each $n\geq 1$, there is a unique matrix $F_{n}$ of positive (i.e., non-negative) integers
such that $\eta_{n}$ is multiplication by $F_{n}$.
Set $f=\left((F_{n})_{n=1}^{\infty},(f_{n})_{n=1}^{\infty}\right)$.
Now it is easy to check  that $f$ is a premorphism in $\mathbf{BD}$ and $\mathcal{D}([f])=\varphi$.
Hence $\mathcal{D}$ is a full functor.

Finally, that $\mathcal{D}$ is essentially surjective follows from
 the Effros-Handelman-Shen theorem
\cite[Theorem~2.2]{ehs80}.
\end{proof}
\begin{proposition}
The following diagram of functors commutes (up to natural isomorphism):

\[
\xymatrix{
\overline{\mathbf{AF}^{\mathrm{out}}}
\ar[r]^{\overline{\mathcal{B}}}\ar[rd]_{\overline{\mathrm{K}}_{0}}
 & \mathbf{BD}
  \ar[d]^{\mathcal{D}}\\
& \mathbf{DG}\ .
}
\]
Moreover, all three functors in this diagram are equivalence of categories.
\end{proposition}
\begin{proof}
By Theorem~\ref{thmd} and \cite[Theorems~5.11 and 6.4]{aeg14},
all three functors in the above diagram are equivalence of categories.
That the diagram commutes (up to natural isomorphism) follows from
the continuity of the functor $\mathrm{K}_{0}$  with respect to  direct limits.
\end{proof}
\section{The Functor $\mathrm{K}^{0}$}
\noindent
In this section we investigate the functor
$\mathrm{K}^{0}:\mathbf{DS}\to \mathbf{DG}_{1}$
from the category of essentially minimal t.d.~dynamical systems on totally disconnected, compact
metrizable spaces (Definition~\ref{defcatds}) to the category of unital dimension groups
(Definition~\ref{defcatdg1}).
Let $X$ be a totally disconnected, compact metrizable   space and let
$\varphi$ be a homeomorphism of $X$. Let $\mathrm{C}(X,\Z)$
denote the group of continuous functions from $X$ to $\Z$. Set
\[
\mathrm{C}(X,\Z)^{+}=\{f\in \mathrm{C}(X,\Z)\mid f\geq 0\}.
\]
Let $\textbf{1}$ be the constant function with value $1$ which is an order unit for
$\mathrm{C}(X,\Z)$. Denote by $\varphi_{*}$ the automorphism of
$\mathrm{C}(X,\Z)$ sending $f$ to $f\circ \varphi^{-1}$. Then
$\mathrm{id}-\varphi_{*}$ is an endomorphism  of
$\mathrm{C}(X,\Z)$, where $\mathrm{id}$ is the identity map. Define
$\mathrm{K}^{0}(X,\varphi)$ as in \cite{hps92}, i.e.,
\[
\mathrm{K}^{0}(X,\varphi)=\frac{\mathrm{C}(X,\Z)}{B_{\varphi}},
\]
where $B_{\varphi}$ denotes the subgroup
${\mathrm{Im}}(\mathrm{id}-\varphi_{*})$ of $\mathrm{C}(X,\Z)$.
\begin{proposition}[\cite{hps92}]
Let $(X,\varphi,y)$ be an essentially minimal t.d.~system. If we define
$\mathrm{K}^{0}(X,\varphi)^{+}$ to be the image of
$\mathrm{C}(X,\Z)^{+}$ in the quotient group
$\mathrm{K}^{0}(X,\varphi)$ and if we also let $\mathbf{1}$ denote the image of $\mathbf{1}$, then
\[
(\mathrm{K}^{0}(X,\varphi),\mathrm{K}^{0}(X,\varphi)^{+})
\]
is a dimension group and $\mathbf{1}$ is an order unit.
\end{proposition}
\begin{definition}\label{defcatdg1}
Denote by $\mathbf{DG}_{1}$ the category whose objects are triples $(G,G^{+},u)$ where
$(G,G^{+})$ is a dimension group and $u$ is a fixed order unit. A morphism in
$\mathbf{DG}_{1}$ is a positive homomorphism preserving the distinguished order units.
Let us refer to $\mathbf{DG}_{1}$ as the category of unital dimension groups.
\end{definition}
\begin{definition}
Define the contravariant functor
$\mathrm{K}^{0}:\mathbf{DS}\to \mathbf{DG}_{1}$ as follows. For each
essentially minimal t.d.~dynamical  system $\mathcal{X}=(X,\varphi,y)$ set
 $\mathrm{K}^{0}(\mathcal{X})=(\mathrm{K}^{0}(X,\varphi),\mathrm{K}^{0}(X,\varphi)^{+},\mathbf{1}) $.
The definition $\mathrm{K}^{0}$ on morphisms proceeds in a natural way.
In fact, let
$\alpha :(X,\varphi,y)\to (Y,\psi,z)$ be a morphism in $\mathbf{DS}$.
Consider the group homomorphism
$\mathrm{C}(\alpha): \mathrm{C}(Y,\Z)\to \mathrm{C}(X,\Z)$ defined by
$\mathrm{C}(\alpha)(f)=f\circ \alpha$. It is easy to see that
$\mathrm{C}(\alpha)(B_{\psi})\subseteq
\mathrm{Im}(B_{\varphi})$.
Set
\[
\mathrm{K}^{0}(\alpha)(f+B_{\psi})=
f\circ \alpha+B_{\varphi}.
\]
\end{definition}

The proof of the following statement is straightforward.
\begin{proposition}
The map
 $\mathrm{K}^{0}:\mathbf{DS}\to \mathbf{DG}_{1}$,
 as defined above, is a contravariant functor.
\end{proposition}
Let us recall the notion of a (strong) classification functor.
\begin{definition}[Elliott, \cite{el10}]\label{defscf}
A functor ${F}:\mathcal{C}\to \mathcal{D}$
is called a \emph{classification functor} if
${F}(a)\cong {F}(a)$ implies $a\cong b$, for each $a,b\in \mathcal{C}$; it is   called a
\emph{strong classification functor} if each isomorphism from
${F}(a)$ to ${F}(b)$ is the image of an isomorphism from $a$ to $b$.
Thus in this case we have $a\cong b$ if and only if ${F}(a)\cong {F}(b)$.
\end{definition}
\begin{remark}
The functor $\mathrm{K}^{0}:\mathbf{DS}\to \mathbf{DG}_{1}$
is not a (strong) classification functor. In fact, in view of Theorem~\ref{thrgps}, we need
a weaker notion of morphism between dynamical systems to obtain a classification functor.
However, the existence of this functor may be important from other points of view.
For example, Glasner and Weiss used this (for the special case of Cantor systems)
 to deal with their weak orbit equivalence
(see \cite[Proposition~3.1]{gw95}).
\end{remark}


\section{The Functor $\mathcal{AF}$ from $\mathbf{DS}$ to $\mathbf{AF}$}
\noindent
In this section we construct the functor $\mathcal{AF}$
from the category of essentially minimal t.d.~dynamical systems (Definition~\ref{defcatds})
to the category of AF~algebras.


Let $(X,\varphi,y)$ be an essentially minimal t.d.~dynamical system (Definition~\ref{defemds}) and
$\mathrm{C}(X)\rtimes_{\varphi}\mathbb{Z}$ denote the associated crossed product C*-algebra
 as described in \cite{hps92}. Set
$\mathrm{C}(X)\rtimes_{\varphi}\mathbb{Z}=\mathcal{A}(X,\varphi,y)$.
In fact, the homeomorphism $\varphi$ of $X$ induces the automorphism
$\mathrm{C}(\varphi^{-1})$ of $\mathrm{C}(X)$ defined by
$\mathrm{C}(\varphi^{-1})(f)=f\circ \varphi^{-1}$, $f\in \mathrm{C}(X)$.
Then $\mathbb{Z}$ acts on $\mathrm{C}(X)$ by means of this automorphism
and one considers the resulting crossed product C*-algebra
$\mathrm{C}(X)\rtimes_{\varphi}\mathbb{Z}$. See \cite{ped79}, \cite[Chapter~VIII]{da96},
and \cite{wil07} for the definition of (discrete) crossed products of C*-algebras.
Also, each morphism in $\mathbf{DS}$ induces a morphism
in the category of unital C*-algebras with unital $*$-homomorphisms,
$\mathcal{C}^{*}_{1}$ (see the proof of the following proposition).
\begin{proposition}\label{propfa}
The map $\mathcal{A}:\mathbf{DS}\to \mathcal{C}^{*}_{1}$, as defined above,
is a contravariant functor.
\end{proposition}
\begin{proof}
Let $(X,\varphi,y)$ and $(Y,\psi,z)$ be in $\mathbf{DS}$ and
$\alpha :(X,\varphi,y)\to (Y,\psi,z)$ be a morphism, i.e.,  a homomorphism from
the dynamical system $(X,\varphi)$ to $(Y,\psi)$ (a continuous map
with $\alpha\circ \varphi=\psi\circ \alpha$) such that $\varphi(y)=z$.
Then $\mathrm{C}(\alpha):\mathrm{C}(Y)\to \mathrm{C}(X)$
defined by $\mathrm{C}(\alpha)(f)=f\circ \alpha$, is an equivariant homomorphism.
In fact, for each $n\in\mathbb{Z}$ and each $f\in\mathrm{C}(Y)$ we have
\[
\mathrm{C}(\alpha)(f\circ \psi^{n})=f\circ \psi^{n}\circ \alpha=f\circ\alpha\circ \varphi^{n}
=\mathrm{C}(\alpha)(f)\circ  \varphi^{n}.
\]
Therefore, by \cite[Corollary~2.48]{wil07}, $\mathrm{C}(\alpha)$ induces a unital $*$-homomorphism
$\mathcal{A}(\alpha): \mathrm{C}(Y)\rtimes_{\psi}\mathbb{Z}\to
\mathrm{C}(X)\rtimes_{\varphi}\mathbb{Z}$ such that
$\mathcal{A}(\alpha)(g)=\mathrm{C}(\alpha)\circ g$, for each $g\in \mathrm{C_{c}}(\mathbb{Z},\mathrm{C}(Y))$.
It is easy to check that the map $\mathcal{A}:\mathbf{DS}\to \mathcal{C}^{*}_{1}$ preserves  identities and
composition and so is
a contravariant functor.
\end{proof}
\begin{remark}
In the proof of the above proposition, we did not use the essentially minimality. In fact, the contravariant functor $\mathcal{A}$ may also be defined
for general dynamical systems.
\end{remark}
\begin{definition}\label{deffaf}
Let $X$ be a totally disconnected metrizable compact space, and
let $\mathcal{X}=(X,\varphi,y)$ be an essentially minimal t.d.~dynamical system.
Denote by $\mathcal{AF}(\mathcal{X})$ the C*-subalgebra of
$\mathcal{A}(\mathcal{X})=\mathrm{C}(X)\rtimes_{\varphi}\mathbb{Z}$ generated by
$\mathrm{C}(X)$ and $u\cdot \mathrm{C}_{0}(X\setminus \{y\})$ where $u$ is the canonical unitary
determining $\varphi$ in $\mathrm{C}(X)\rtimes_{\varphi}\mathbb{Z}$.
By \cite[Theorem~3.3]{pu89}, $\mathcal{AF}(\mathcal{X})$ is an AF~algebra.
\end{definition}
\begin{remark}
The definition of an AF~algebra associated with an essentially minimal t.d.~dynamical system
$(X,\varphi,y)$ has been taken from \cite{pu89, hps92}. In \cite{hps92}, the authors
 use the notation $AF(X,\varphi,y)$. Also, for each closed subset $Z$ of $X$, the
 C*-subalgebra of
$\mathcal{A}(\mathcal{X})=\mathrm{C}(X)\rtimes_{\varphi}\mathbb{Z}$ generated by
$\mathrm{C}(X)$ and $u\cdot \mathrm{C}_{0}(X\setminus Z)$ is an AF~algebra \cite{pu89}.
\end{remark}
The functor $\mathcal{A}$ in Proposition~\ref{propfa} preserves the AF~algebra parts and
hence we obtain a functor $\mathcal{AF}$ from $\mathbf{DS}$ to the category of
AF~algebras, $\mathbf{AF}$ (see the proof of the following proposition).
\begin{proposition}\label{propfuncaf}
The map $\mathcal{AF}:\mathbf{DS}\to \mathbf{AF}$, as defined above,
is a contravariant functor.
\end{proposition}
\begin{proof}
Let us examine the definition of the map $\mathcal{AF}$ on morphisms. Let
$\mathcal{X}_{1}=(X,\varphi,y)$ and $\mathcal{X}_{2}=(Y,\psi,z)$ be in
$\mathbf{DS}$ and $\alpha : \mathcal{X}_{1} \to \mathcal{X}_{2}$ be a morphism in
$\mathbf{DS}$. Consider the $*$-homomorphism
$\mathcal{A}(\alpha): \mathcal{A}(\mathcal{X}_{2})\to \mathcal{A}(\mathcal{X}_{1})$
as in Proposition~\ref{propfa}. We show that
$\mathcal{A}(\alpha)(\mathcal{AF}(\mathcal{X}_{2}))\subseteq \mathcal{AF}(\mathcal{X}_{1})$.
It is obvious that
$\mathcal{A}(\alpha)(\mathrm{C}(Y))\subseteq \mathrm{C}(X)$.
Denote by $u_{1}$ and $u_{2}$ the canonical unitaries determining	
$\varphi$ and $\psi$ in
$\mathrm{C}(X)\rtimes_{\varphi}\mathbb{Z}$ and
$\mathrm{C}(Y)\rtimes_{\psi}\mathbb{Z}$, respectively.
Then we have
$\mathcal{A}(\alpha)(u_{2})=u_{1}$.
Now let $f\in \mathrm{C}_{0}(Y\setminus \{z\})$. Then,
\[\mathcal{A}(\alpha)(f)(y)=(f\circ \alpha) (y)= f(\alpha(y))=f(z)=0.\]
This shows that
$\mathcal{A}(\alpha)(\mathrm{C}_{0}(Y\setminus \{z\}))\subseteq \mathrm{C}_{0}(X\setminus \{y\})$.
Therefore, in view of Definition~\ref{deffaf},
$\mathcal{A}(\alpha)(\mathcal{AF}(\mathcal{X}_{2}))\subseteq \mathcal{AF}(\mathcal{X}_{1})$.
Denote by
$\mathcal{AF}(\alpha):\mathcal{AF}(\mathcal{X}_{2})\to \mathcal{AF}(\mathcal{X}_{1})$
 the restriction of $\mathcal{A}(\alpha)$ to $\mathcal{AF}(\mathcal{X}_{2})$.
By Proposition~\ref{propfa}, $\mathcal{AF}:\mathbf{DS}\to \mathbf{AF}$ thus defined
is a contravariant functor.
\end{proof}
\begin{remark}
The functor $\mathcal{AF}:\mathbf{DS}\to \mathbf{AF}$ is faithful. This is
because for each morphism
$\alpha : \mathcal{X}_{1} \to \mathcal{X}_{2}$   as in the proof of Proposition~\ref{propfuncaf}, we have
$\mathcal{AF}(\alpha)(\mathrm{C}(Y))=\mathcal{A}(\alpha)(\mathrm{C}(Y))\subseteq \mathrm{C}(X)$
and
$\mathcal{AF}(\alpha)(f)=f\circ \alpha$, $f\in\mathrm{C}(Y)$.
 The functor $\mathcal{AF}$ is not a strong classification functor
(by Theorem~\ref{thrgps} below). Therefore,  $\mathcal{AF}$ is not a full functor,
since each full and faithful functor is a a strong classification functor
(by \cite[Lemma~5.10]{aeg14}). Similarly,
 the functor
 $\mathcal{A}:\mathbf{DS}\to \mathcal{C}^{*}_{1}$ is faithful but not full. Also, it not a
 strong classification functor.
\end{remark}
 That the first three statements in the following theorem
 are equivalent is a well-known result \cite{gps95}.
Recall that a minimal dynamical system $(X,\varphi)$ is called a \emph{Cantor minimal system}
if $X$ is a compact metrizable
space with a countable basis of clopen subsets  and
$X$ has no isolated points (i.e., $X$ is homeomorphic to the Cantor set).
See \cite{gps95} for the definition of the notion of strong orbit equivalence.
\begin{theorem}\label{thrgps}
Let $(X,\varphi)$ and $(Y,\psi)$ be Cantor minimal systems. Let $y$ and $z$ be arbitrary points in
$X$ and $Y$, respectively. Then the following statements are equivalent:
\begin{itemize}
\item[(1)]
$(X,\varphi)$ and $(Y,\psi)$ are strong orbit equivalent;
\item[(2)]
$\mathrm{K}^{0}(X,\varphi)$ is order isomorphic to $\mathrm{K}^{0}(Y,\psi)$ by a
map preserving the distinguished order unit;
\item[(3)]
$\mathrm{C}(X)\rtimes_{\varphi}\mathbb{Z}\cong \mathrm{C}(Y)\rtimes_{\psi}\mathbb{Z}$;
\item[(4)]
$\mathcal{AF}(X,\varphi,y)\cong \mathcal{AF}(Y,\psi,z)$ in $\mathbf{AF}$.
\end{itemize}
\end{theorem}
\begin{proof}
The theorem follows from \cite[Theorem~2.1]{gps95} and \cite[Theorem~5.3]{hps92}.
\end{proof}
\begin{remark}
One way to obtain a functorial formulation of the preceding theorem is to define a suitable
notion of morphism between Cantor minimal systems such that isomorphism coincides with
 strong orbit equivalence as introduced in \cite{gps95}.
This needs more investigation.
\end{remark}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Refrences
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{99}
\bibitem{aeg14} M. Amini, G.\,A. Elliott, and N. Golestani, \emph{The category of Bratteli diagrams}, 
Canad. J. Math. \textbf{67} (2015), 990--1023.
\bibitem{br72} O. Bratteli, \emph{Inductive limits of finite
dimensional C$^{*}$-algebras},  Trans. Amer. Math. Soc. \textbf{171} (1972),
195--234.
\bibitem{da96} K.\,R. Davidson, \emph{C$^{*}$-Algebras by Example}, Fields Institute Monographs 6, American Mathematical Society, Providence, 1996.
\bibitem{dhs99} F. Durand, B. Host, and C.\,F. Skau,  \emph{Substitutional dynamical systems,
Bratteli diagrams and dimension groups}, Ergodic Theory Dynam.
Systems \textbf{19} (1999), no. 4,  953--993.
\bibitem{ef81} E.\,G. Effros,  \emph{Dimensions and C*-algebras}, Conference Board Math. Sci.
 \textbf{46} Amer. Math. Soc., Providence, R.I., 1981.
\bibitem{ehs80} E.\,G. Effros, D.\,E. Handelman, and C.\,L. Shen, \emph{Dimension groups and their affine transformations}, Amer. J. Math. \textbf{102} (1980), 385--402.
\bibitem{el76} G.\,A. Elliott, \emph{On the classification of inductive limits of sequences of semisimple finite dimensional algebras}, J. Algebra \textbf{38} (1976), 29--44.
\bibitem{el10} G.\,A. Elliott, \emph{Towards a theory of classification}, Adv. Math. \textbf{223} (2010),  30--48.
\bibitem{gps95} R. Giordano, I.\,F. Putnam, and C.\,F. Skau, \emph{Topological orbit equivalence
and $\mathrm{C}^{*}$-crossed products}, J. Reine Angew. Math. \textbf{469} (1995), 51--111.
\bibitem{gw95} E. Glasner and B. Weiss, \emph{Weak orbit equivalence of Cantor minimal systems},
Int. J. Math.  \textbf{6} (1995), 559--579.
\bibitem{gb84} R. Goldblatt, \emph{Topoi, the Categorical  Analysis of
    Logic}, Elsevier Science Publishers, New York, 1984.
\bibitem{hps92} R.\,H. Herman, I.\,F. Putnam, and C.\,F. Skau, \emph{Ordered Bratteli diagrams,
 dimension groups and topological dynamics},  Internat. J. Math. \textbf{3} (1992), no. 6, 827--864.
%\bibitem{lt80} A.\,J. Lazar, D.\,C. Taylor, \emph{Approximately finite-dimensional
%C$^{*}$-algebras and Bratteli diagrams}, Trans. Amer. Math. Soc. \textbf{259}
%(1980), 599--619.
\bibitem{ma98} S. Mac Lane, \emph{Categories for the  Working Mathematician},
Second edition, Springer, New York, 1998.
\bibitem{ma02} H. Matui,  \emph{Topological orbit equivalence of locally compact Cantor minimal systems}, Ergodic theory and dynam. systems \textbf{22} (2002), no. 06, 1871--1903.
%\bibitem{ta79} M. Takesaki, \emph{Theory of Operator Algebras, vol. I}, Springer, New York, 1979.
\bibitem{ped79} Gert K. Pedersen, \emph{C*-Algebras and Their Automorphism
Groups}, London Mathematical  Society Monographs, Vol. 14, Academic Press, London, 1979.
\bibitem{po91} S.\,C. Power, \emph{Algebraic orders on $K_0$ and approximately finite operator algebras},
arXiv:funct-an/9302002.
\bibitem{pu89} I.\,F. Putnam, \emph{C*-algebras associated with minimal homeomorphisms of the Cantor
set}, Pacific J. Math. \textbf{136} (1989),  329--353.
\bibitem{ve81a} A.\,V. Ver\v{s}ik, \emph{A theorem on periodical Markov approximation in ergodic theory},
Zapiski Nauchn. Semin. LOMI \textbf{115} (1982), 72--82.
English translation: Ergodic theory and related topics (Vitte, 1981), Math. Res. \textbf{12},
Akademie-Verlag, Berlin,  195--206, 1982.
\bibitem{ve81b} A.\,V. Ver\v{s}ik, \emph{Uniform algebraic approximations of shift and multiplication operators},
Dokl. Akad. Nauk SSSR \textbf{259} (1981), no.3, 526--529. English translation:
Sov. Math. Dokl. \textbf{24} (1981), 97--100.

\bibitem{wo93} N.\,E. Wegge-Olsen, \emph{K-Theory and
C$^{*}$-Algebras}, The Clarendon Press, New York, 1993.
\bibitem{wil07} D.\,P. Williams, \emph{Crossed Products of C*-Algebras}, Vol. 134,
American Mathematical Society, 2007.
 \end{thebibliography}
\end{document}

1509.07246.extracted_bib
{aeg14 M. Amini, G.\,A. Elliott, and N. Golestani, The category of Bratteli diagrams, Canad. J. Math. 67 (2015), 990--1023.
1509.07246.extracted_bib
{br72 O. Bratteli, Inductive limits of finite dimensional C$^{*$-algebras, Trans. Amer. Math. Soc. 171 (1972), 195--234.
1509.07246.extracted_bib
{da96 K.\,R. Davidson, C$^{*$-Algebras by Example, Fields Institute Monographs 6, American Mathematical Society, Providence, 1996.
1509.07246.extracted_bib
{dhs99 F. Durand, B. Host, and C.\,F. Skau, Substitutional dynamical systems, Bratteli diagrams and dimension groups, Ergodic Theory Dynam. Systems 19 (1999), no. 4, 953--993.
1509.07246.extracted_bib
{ef81 E.\,G. Effros, Dimensions and C*-algebras, Conference Board Math. Sci. 46 Amer. Math. Soc., Providence, R.I., 1981.
1509.07246.extracted_bib
{ehs80 E.\,G. Effros, D.\,E. Handelman, and C.\,L. Shen, Dimension groups and their affine transformations, Amer. J. Math. 102 (1980), 385--402.
1509.07246.extracted_bib
{el76 G.\,A. Elliott, On the classification of inductive limits of sequences of semisimple finite dimensional algebras, J. Algebra 38 (1976), 29--44.
1509.07246.extracted_bib
{el10 G.\,A. Elliott, Towards a theory of classification, Adv. Math. 223 (2010), 30--48.
1509.07246.extracted_bib
{gps95 R. Giordano, I.\,F. Putnam, and C.\,F. Skau, Topological orbit equivalence and $\mathrm{C^{*$-crossed products, J. Reine Angew. Math. 469 (1995), 51--111.
1509.07246.extracted_bib
{gw95 E. Glasner and B. Weiss, Weak orbit equivalence of Cantor minimal systems, Int. J. Math. 6 (1995), 559--579.
1509.07246.extracted_bib
{gb84 R. Goldblatt, Topoi, the Categorical Analysis of Logic, Elsevier Science Publishers, New York, 1984.
1509.07246.extracted_bib
{hps92 R.\,H. Herman, I.\,F. Putnam, and C.\,F. Skau, Ordered Bratteli diagrams, dimension groups and topological dynamics, Internat. J. Math. 3 (1992), no. 6, 827--864. %
1509.07246.extracted_bib
{lt80 A.\,J. Lazar, D.\,C. Taylor, Approximately finite-dimensional %C$^{*$-algebras and Bratteli diagrams, Trans. Amer. Math. Soc. 259 %(1980), 599--619.
1509.07246.extracted_bib
{ma98 S. Mac Lane, Categories for the Working Mathematician, Second edition, Springer, New York, 1998.
1509.07246.extracted_bib
{ma02 H. Matui, Topological orbit equivalence of locally compact Cantor minimal systems, Ergodic theory and dynam. systems 22 (2002), no. 06, 1871--1903. %
1509.07246.extracted_bib
{ta79 M. Takesaki, Theory of Operator Algebras, vol. I, Springer, New York, 1979.
1509.07246.extracted_bib
{ped79 Gert K. Pedersen, C*-Algebras and Their Automorphism Groups, London Mathematical Society Monographs, Vol. 14, Academic Press, London, 1979.
1509.07246.extracted_bib
{po91 S.\,C. Power, Algebraic orders on $K_0$ and approximately finite operator algebras, arXiv:funct-an/9302002.
1509.07246.extracted_bib
{pu89 I.\,F. Putnam, C*-algebras associated with minimal homeomorphisms of the Cantor set, Pacific J. Math. 136 (1989), 329--353.
1509.07246.extracted_bib
{ve81a A.\,V. Ver\v{sik, A theorem on periodical Markov approximation in ergodic theory, Zapiski Nauchn. Semin. LOMI 115 (1982), 72--82. English translation: Ergodic theory and related topics (Vitte, 1981), Math. Res. 12, Akademie-Verlag, Berlin, 195--206, 1982.
1509.07246.extracted_bib
{ve81b A.\,V. Ver\v{sik, Uniform algebraic approximations of shift and multiplication operators, Dokl. Akad. Nauk SSSR 259 (1981), no.3, 526--529. English translation: Sov. Math. Dokl. 24 (1981), 97--100.
1509.07246.extracted_bib
{wo93 N.\,E. Wegge-Olsen, K-Theory and C$^{*$-Algebras, The Clarendon Press, New York, 1993.
1509.07246.extracted_bib
{wil07 D.\,P. Williams, Crossed Products of C*-Algebras, Vol. 134, American Mathematical Society, 2007.