text:
\section{Introduction and main results}

In 1975, graph inverse semigroups were first introduced by Ash and Hall in \cite{AshHall} where they gave necessary and sufficient conditions for the semigroups to be congruence free. Such semigroups are defined based on special kinds of directed graphs where there exists at most one edge from one vertex to another. The definition of graph inverse semigroups based on general directed graphs can be found in \cite{Jones, JonesLawson, MesyanMitchell}. These inverse semigroups are related to various types of algebras, for instance, $C^\ast$-algebras, Cohn path algebras, Leavitt path algebras and so on (see \cite{AbramsPino, AlaliGilbert, JonesLawson, Krieger, KumjianPaskRaeburnRenault, KumjianPaskRaeburn, Paterson}). Graph inverse semigroups have also been studied in their own right in recent years (see \cite{AlaliGilbert, ChihPlessas, Jones, JonesLawson, MesyanMitchell, MesyanMitchellMoraynePeresse}).

A {\it directed graph} $G = (V, E, r, s)$ consists of $V, E$ and functions $r, s : E \rightarrow V$. The elements of $V$ are called {\it vertices} and the elements of $E$ are called {\it edges}. For each edge $e$, $s(e)$ is the {\it source} of $e$ and $r(e)$ is the {\it range} of $e$. The cardinality $|s^{-1}(v)|$ of a vertex $v$ is called the {\it index} of $v$. In this paper, we shall refer to a directed graphs simply as a ``graph''.

Now let $G = (V, E, r, s)$ be a graph. The {\it graph inverse semigroup} $\In$ of $G$ is the semigroup with zero generated by $V$ and $E$, together with the set $E^\ast$ of variables $\{e^\ast : e \in E\}$, satisfying the following relations for all $u, v \in V$ and $e, f \in E$:
\begin{description}
\item[{\rm(GI1)}] $u v = \delta_{u,v} u$;
\item[{\rm(GI2)}] $s(e) e = e r(e) = e$;
\item[{\rm(GI3)}] $r(e) e^\ast = e^\ast s(e) = e^\ast$;
\item[{\rm(CK1)}] $f^\ast e = \delta_{f,e} r(e)$.
\end{description}
A {\it path} in $G$ is a sequence $\al = e_1 \cdots e_n$ of edges such that $r(e_i) = s(e_{i+1})$ for $i = 1, \cdots, n-1$. In such a case, $s(\al) := s(e_1)$ is the {\it source} of $\al$, $r(\al) := r(e_n)$ is the {\it range} of $\al$, $|\al| := n$ is the {\it length} of $\al$ and $\al^\ast := e_n^\ast \cdots e_1^\ast$ the {\it ghost path} of $\al$. We view an element $v$ of $V$ as a path of length $0$, $s(v) = r(v) = v$ and $v^\ast = v$. Denote as $\pg$ the set of all paths in $G$. Every nonzero element in $\In$ can be uniquely written as $\al\be^\ast$ for some $\al, \be \in \pg$ with $r(\al) = r(\be)$. One can also verify that $\In$ is an inverse semigroup with zero such that $(\al\be^\ast)^{-1} = \be\al^\ast$ for any $\al, \be \in \pg$. The idempotent set $E(\In)$, which is a (meet) semilattice, consists of all elements $\al\al^\ast$ with $\al \in \pg$. Moreover, for any $\al \be^\ast, \ze \eta^\ast \in \In$, we have
\begin{equation} \label{multiplication}
(\al \be^\ast) (\ze \eta^\ast) =
\left\{ \begin{array}{ll}
(\al\xi)\eta^\ast & \mbox{if $\ze = \be\xi$ for some $\xi \in \pg$} \\
\al(\eta\xi)^\ast & \mbox{if $\be = \ze\xi$ for some $\xi \in \pg$} \\
0 & \mbox{otherwise}.
\end{array}\right.
\end{equation}

In \cite{MesyanMitchell}, Mesyan and Mitchell showed that the quotient of any graph inverse semigroup by a Rees congruence is always isomorphic to another graph inverse semigroup. They also found a graph theoretical condition for a not necessarily Rees congruence and a graph theoretical condition for a graph inverse semigroup to have only Rees congruences. Jones also investigated non-Rees congruences on certain kinds of graph inverse semigroups (see \cite{Jones}).

In this paper we give a description of congruences on a graph inverse semigroup in terms of the underlying graph that is different from \cite{MesyanMitchell}.  This description implies that the inverse graph semigroup of a finite graph is congruence Noetherian.

To present our main result, we need some more notions and notation. {\it In what follows, $G = (V, E, r, s)$ is always a graph and $\In$ is the graph inverse semigroup of $G$}.

A subset $H$ of $V$ is {\it hereditary} if $s(e) \in H$ always implies $r(e) \in H$ for $e \in E$. Let $H$ be a hereditary subset of $V$, $V_1 = V \setminus H$, $E_1 = \{e \in E : r(e) \not\in H\}$, $s_{G \setminus H} = s |_{E_1}$ and $r_{G \setminus H} = r |_{E_1}$. Then we denote the graph $(V_1, E_1, s_{G \setminus H}, r_{G \setminus H})$ as $G \setminus H$.

Let $\rho$ be a congruence on $\In$. Then clearly $H = 0 \rho \cap V$ is a hereditary subset of $V$. Passing to the graph $G \setminus H$, we shall assume without loss of generality that $H = \emptyset$. So for graph inverse semigroups, we only need to describe these {\it special congruences} where no vertex is equivalent to 0.

Let $\al = e_1 \cdots e_n$ be a path in $G$ and $v(\al)$ denote the set $\{s(e_i) : i = 1, \cdots, n\}$. If $| \al | = 0$, we appoint that $v(\al) = \emptyset$. An edge $e$ is an {\it exit} to the path $\al$ if there exists $i$ such that $s(e) = s(e_i)$ and $e \neq e_i$. The path $\al$ is said to be {\it no-exit} if it does not have an exit. The path $\al$ is called a {\it cycle} if $s(\al) = r(\al)$ and $s(e_i) \neq s(e_j)$ whenever $i \neq j$. For any cycle $c$, let $\cir c$ be the set of all cycles that are obtained from $c$ by cyclic permutations. Two cycles $c_1$ and $c_2$ are said to be {\it disjoint} if $v(c_1) \cap v(c_2) = \emptyset$.

We write $\bar{V} := \{v \in V : | s^{-1}(v) | = 1\}$, the set of vertices of index one. For any $v \in \bar{V}$, we denote the unique edge in $s^{-1}(v)$ as $e_v$. Let $W$ be a subset of $\bar{V}$, $\mathbb{Z}^+$ be the set of all positive integers and $\cnw$ be the set of all cycles whose vertices lie in $W$. Since all vertices in $W$ have index one, an arbitrary cycle $c \in \cnw$ is no-exit and any two cycles in $\cnw$ are either disjoint or cyclic permutations of each other.

\begin{defn} \label{Pair}
A {\it cycle function} $f : \cnw \rightarrow \mathbb{Z}^+ \cup \{\infty\}$ is a function that is invariant under cyclic permutations. A {\it congruence pair $(W, f)$ of $G$} consists of a subset $W$ of $\bar{V}$ and a cycle function $f$.
\end{defn}

\begin{defn} \label{Triple}
Let $H$ be a hereditary subset of $V$ and $(W, f)$ be a congruence pair of $G \setminus H$. Then $(H, W, f)$ is called a {\it congruence triple of $G$}.
\end{defn}

Let us show how a congruence gives rise to a congruence triple (respectively, a special congruence gives rise to a congruence pair).

Let $\rho$ be a congruence on $\In$, $H = 0 \rho \cap V$ and $W = \{v \in V \setminus H : e  e^\ast \rho v \mbox{ for } e \in s^{-1}_{G \setminus H} (v)\}$. Later we shall see from Lemma~\ref{NoExit} that in this case all vertices in $W$ have index one in $G \setminus H$. Let $\cnw$ consist of all distinct cycles whose vertices lie in $W$. For $c \in \cnw$, let $f(c)$ be the smallest positive integer $m$ such that $c^m \rho s(c)$. If no power of $c$ is equivalent to $s(c)$, then we define $f(c) = \infty$. Then $T(\rho) = (H, W, f)$ is a congruence triple.

If $\rho$ is a special congruence, then $T(\rho) = (\emptyset, W, f)$. In this case, $(W, f)$ is a congruence pair.

Conversely, let $(W, f)$ be a congruence pair of $G$ and let $\wp(W, f)$ denote the congruence generated by $\mathbf{R}$ consisting of all pairs $(e_v e_v^\ast, v)$ for $v \in W$ and $(c^{f(c)}, s(c))$ for $c \in \cnw$ with $f(c) \in \mathbb{Z}^+$.

Let $(H, W, f)$ be a congruence triple of $G$ and let $\wp(H, W, f)$ denote the congruence generated by all pairs $(v, 0)$ for $v \in H$, $(ee^\ast, w)$ for $w \in W$ with $s(e) = w$ and $(c^{f(c)}, s(c))$ for $c \in \cnw$ with $f(c) \in \mathbb{Z}^+$.

\begin{thm} \label{Main1}
The mapping $T$ from the set of all special congruences on $\In$ to the set of all congruence pairs of $G$ and the mapping $\wp$ from the set of all congruence pairs of $G$ to the set of all special congruences on $\In$ are inverses. In particular, there exists a one-to-one correspondence between special congruences on $\In$ and congruence pairs of $G$.
\end{thm}

Theorem~\ref{Main1} immediately implies

\begin{thm} \label{Main2}
The mapping $T$ from the set of all congruences on $\In$ to the set of all congruence triples of $G$ and the mapping $\wp$ from the set of all congruence triples of $G$ to the set of all congruences on $\In$ are inverses. In particular, there exists a one-to-one correspondence between congruences on $\In$ and congruence triples of $G$.
\end{thm}

A graph inverse semigroup is said to be {\it congruence Noetherian} if for any congruence sequence
$$
\rho_1 \subseteq \rho_2 \subseteq \cdots \subseteq \rho_n \subseteq \cdots,
$$
there exists $m \in \mathbb{Z}^+$ such that $\rho_m = \rho_{m+k}$ for all $k \in \mathbb{Z}^+$.

\begin{cor} \label{Noetherian}
Each graph inverse semigroup of a finite graph is congruence Noetherian.
\end{cor}

\section{Proof of the main results}

We need to prove Theorem~\ref{Main1} and Corollary~\ref{Noetherian}. Let $\mathbb{N}$ be the set of all nonnegative integers. For any cycle $c$ and any positive integer $m$, we appoint that $c^0 = s(c)$ and $(c^\ast)^m = c^{-m}$. The next lemma directly follows from the related definitions.

\begin{lem} \label{CircleAdd}
For any no-exit cycle $c$ and $\al \in \pg$ with $s(c) = s(\al)$, there exist $\al_1, \al_2 \in \pg$ and $k \in \mathbb{N}$ such that $\al = c^k \al_1$, $c = \al_1 \al_2$ and $|\al_1| < |c|$.
\end{lem}

\begin{lem} \label{Circle}
For any no-exit cycle $c$, $\al \in \pg$ and $k \in \mathbb{Z}^+$, if $s(c) = s(\al)$, then there exists $c_1 \in \cir c$ such that
\begin{description}
\item[{\rm(1)}] $\al^\ast c^k \al = c_1^k$, and
\item[{\rm(2)}] $c^k \al \al^\ast = \al c_1^k \al^\ast$.
\end{description}
\end{lem}

\begin{prf}
We see from Lemma~\ref{CircleAdd} that $\al = c^l \al_1$ where $l \in \mathbb{N}$ and $\al_1 \in \pg$ such that $c = \al_1 \al_2$ for some $\al_2 \in \pg$. Obviously, $c_1 = \al_2 \al_1 \in \cir c$. It follows from (CK1) that
\begin{align*}
& \al^\ast c^k \al = \al_1^\ast (c^\ast)^l c^k c^l \al_1 = \al_1^\ast c^k \al_1 = \al_2 c^{k-1} \al_1 = c_1^k, \mbox{ and} \\
& c^k \al \al^\ast = c^k c^l \al_1 \al^\ast = c^l \al_1 c_1^k \al^\ast = \al c_1^k \al^\ast.
\end{align*}
That is, (1) and (2) hold.
\end{prf}
\medskip

Now let $\rho$ be a special congruence on $\In$ and $T(\rho) = (W, f)$. We shall show that $\rho = \wp (W, f)$.
\bigskip

The next Lemma directly follows from (GI1), (CK1) and the definition of a congruence on a semigroup.

\begin{lem} \label{Basic}
Let $u, v \in V$ and $\al, \be, \ga \in \pg$.
\begin{description}
\item[{\rm(1)}] If $u \neq v$, then $(u, v) \not\in \rho$;
\item[{\rm(2)}] if $(\al \be^\ast, v) \in \rho$, then $s(\al) = s(\be) = v$;
\item[{\rm(3)}] $(\al \be, s(\al)) \in \rho$ if and only if $(\be \al, s(\be)) \in \rho$;
\item[{\rm(4)}] if $(\al \be \ga^\ast \al^\ast, s(\al)) \in \rho$, then $(\be \ga^\ast, s(\be) \in \rho$;
\item[{\rm(5)}] if $(\al \al^\ast, s(\al)) \in \rho$, $(\be \ga^\ast, s(\be)) \in \rho$ and $r(\al) = s(\be)$, then $(\al \be \ga^\ast \al^\ast, s(\al)) \in \rho$.
\end{description}
\end{lem}

\begin{lem} \label{NoExit}
For any $\al \be^\ast \in \In$, if $(\al \be^\ast, s(\al)) \in \rho$, then both $\al$ and $\be$ are no-exit.
\end{lem}

\begin{prf}
Suppose that $\al = e_1 \cdots e_n$ and $e$ is an exit to $\al$ with $s(e) = s(e_i)$. It follows from Lemma~\ref{Basic}(2) and (CK1) that
$$
(e_i \cdots e_n \be^\ast e_1 \cdots e_{i-1}, s(e_i)) \in \rho.
$$
We obtain from (GI3) and (CK1) that
$$
(0, r(e)) = (0, e^\ast e) = (e^\ast e_i \cdots e_n \be^\ast e_1 \cdots e_{i-1} e, e^\ast s(e_i) e) \in \rho
$$
which contradicts to the hypothesis $0\rho = \{0\}$. Hence, $\al$ is no-exit. We see that $\be$ is also no-exit since $(\al \be^\ast, s(\al)) \in \rho$ implies that $(\be \al^\ast, s(\be) \in \rho$.
\end{prf}

\begin{lem} \label{Subpath}
For any  $\al \in \pg$, if $(\al \al^\ast, s(\al)) \in \rho$ and $\al = \xi \be \eta$ for some $\xi, \be, \eta \in \pg$, then $(\be \be^\ast, s(\be)) \in \rho$.
\end{lem}

\begin{prf}
We see from the hypothesis that $(\xi \be \eta \eta^\ast \be^\ast \xi^\ast, s(\xi)) \in \rho$. Then it follows from  Lemma~\ref{Basic}(4) that $(\be \eta \eta^\ast \be^\ast, s(\be)) \in \rho$ and $(\eta \eta^\ast, s(\eta)) \in \rho$. We see from $(\eta \eta^\ast, s(\eta)) \in \rho$ and the compatibility of $\rho$ that $(\be \eta \eta^\ast \be^\ast, \be \be^\ast) \in \rho$. Hence, we obtain by the transitivity of $\rho$ that $(\be \be^\ast, s(\be)) \in \rho$.
\end{prf}  \medskip

For any $c \in \cnw$, the semigroup $\langle c, c^\ast \rangle$ generated by $c$ and $c^\ast$ is the (infinite) bicyclic semigroup (see \cite{Howie}), and $\langle c, c^\ast \rangle / \rho$ is either the bicyclic semigroup or a cyclic group of order $f(c)$. This implies

\begin{lem} \label{Divisible}
For any $c \in \cnw$, if $f(c) \in \mathbb{Z}^+$, then for every $m \in \mathbb{Z}^+$, $(c^m, s(c)) \in \rho$ implies that $f(c) \mid m$.
\end{lem}

A {\it closed path based at $v$} is a path $\al = e_1 \cdots e_n$ such that $s(\al) = r(\al) = v$. A closed path $\al =  e_1 \cdots e_n$ based at $v$ is called a {\it closed simple path based at $v$} if $s(e_i) \neq v$ for every $i > 1$. A cycle is a closed simple path.

\begin{lem}[{\cite[Lemma~2.3]{AbramsPino}}] \label{CSP}
For any $v \in V$ and  any closed path $\al$ based at $v$, there exist unique closed simple paths $\eta_1, \cdots, \eta_m$ based at $v$ such that $\al = \eta_1 \cdots \eta_m$.
\end{lem}

\begin{lem} \label{PowerofCycle}
If $\al \in \pg \setminus V$ and $(\al, s(\al)) \in \rho$, then there exists a no-exit cycle $c$ such that $\al = c^m$ for some positive integer $m$.
\end{lem}

\begin{prf}
We see from Lemma~\ref{Basic}(2) that $\al$ is a closed path based at $s(\al)$. It follows from Lemma~\ref{CSP} that there exist unique closed simple paths $\mu_1, \cdots, \mu_m$ based at $s(\al)$ such that $\al = \mu_1 \cdots \mu_m$. Then we obtain from Lemma~\ref{NoExit} that $\mu_1 = \cdots = \mu_m = c$ for some no-exit cycle $c$.
\end{prf}  \medskip

The following lemma follows from \cite[Proposition~8]{MesyanMitchell}.

\begin{lem} \label{Classofv}
For any $v \in V$, every element in $v \rho$ is of the form $\al \eta \al^\ast$ or $\al \eta^\ast \al^\ast$ for some $\al, \eta \in \pg$ such that $s(\al) = v$ and $r(\al) = s(\eta) = r(\eta)$.
\end{lem}

\begin{lem} \label{Elementofvrho}
For any $v \in V$, every element in $v \rho$ is of the form either $\al \al^\ast$ or $\al c^{mf(c)} \al^\ast$, where $\al \in \pg$, $s(\al) = v$, $v(\al) \subseteq W$ and in the second case, $m \in \mathbb{Z} \setminus \{0\}$, $r(\al) = s(c)$, $c \in \cnw$ with $f(c) \in \mathbb{Z}^+$.
\end{lem}

\begin{prf}
Note Lemma~\ref{Classofv} and take $x$ in $v \rho$. Then either $x = \al \eta \al^\ast$ or $x = \al \eta^\ast \al^\ast$. We already have $r(\al) = s(\eta)$, $s(\al) = v$ and $v(\al) \subseteq W$ by Lemma~\ref{NoExit}. If $|\eta| = 0$, then it is clear $x = \al \al^\ast$. If $|\eta| > 0$, then noticing $\eta = \eta (s(\eta))^\ast$, we see from Lemmas~\ref{Basic}(4), (2), \ref{PowerofCycle} and \ref{Divisible} that $x = \al c^{m f(c)} \al^\ast$ where $m \in \mathbb{Z} \setminus \{0\}$ and $c \in \cnw$ with $f(c) \in \mathbb{Z}^+$.
\end{prf}

\begin{prop} \label{CPC}
$\wp(W, f) = \rho$.
\end{prop}

\begin{prf}
Recall that $\mathbf{R}$ consists of all pairs $(e_v e_v^\ast, v)$ for $v \in W$ and $(c^{f(c)}, s(c))$ for $c \in \cnw$ with $f(c) \in \mathbb{Z}^+$. Clearly, $\mathbf{R} \subseteq \rho$ so that $\wp(W, f) \subseteq \rho$. For any $\al \be^\ast, \xi \eta^\ast \in \In \setminus \{0\}$, if $(\al \be^\ast, \xi \eta^\ast) \in \rho$, we may suppose $|\al| \leq |\xi|$ without loss of generality. Then we see from Lemma~\ref{NoExit} that $\xi = \al \xi_1$ for some $\xi_1 \in \pg$. Concerning $\be$ and $\eta$, we have the following two cases. \medskip \\
\textbf{Case 1.} $\eta = \be \eta_1$ for some $\eta_1 \in \pg$. It follows from $(\al \be^\ast, \xi \eta^\ast) \in \rho$ and (CK1) that $(r(\al), \xi_1 \eta_1^\ast) \in \rho$. Then we see from Lemmas~\ref{Elementofvrho} and \ref{Subpath} that $(r(\al), \xi_1 \eta_1^\ast) \in \wp(W, f)$. Hence, we have $(\al\be^\ast, \xi \eta^\ast) = (\al \be^\ast, \al \xi_1 \eta_1^\ast \be^\ast) \in \wp(W, f)$. \medskip \\
\textbf{Case 2.} $\be = \eta \be_1$ for some $\be_1 \in \pg \setminus V$. It again follows from $(\al \be^\ast, \xi \eta^\ast) \in \rho$ and (CK1) that $(r(\al), \xi_1 \be_1) \in \rho$. Then we observe from Lemma~\ref{PowerofCycle} that there exists $c \in \cnw$ with $f(c) \in \mathbb{Z}^+$ and such that $\xi_1 \be_1 = c^m$ for some $m \in \mathbb{Z}^+$. Thus, it follows from Lemma~\ref{Divisible} that there exists $k \in \mathbb{Z}^+$ such that $m = k f(c)$. Now we get from the definition of $\wp(W, f)$ that $(c^{kf(c)}, s(c)) \in \wp(W, f)$ which means that $(r(\al), \xi_1 \be_1) \in \wp(W, f)$. Noticing (CK1), we have
$$
(\xi_1^\ast, \be_1) \in \wp(W, f) \Longrightarrow (\be_1^\ast, \xi_1) \in \wp(W, f) \Longrightarrow (\al \be_1^\ast \eta^\ast, \al \xi_1 \eta^\ast) \in \wp(W, f).
$$
That is, $(\al \be^\ast, \xi \eta^\ast) \in \wp(W, f)$. \medskip \\
As a consequence, we obtain that $\rho \subseteq \wp(W, f)$ which leads to $\rho = \wp(W, f)$.
\end{prf}  \medskip

Next let $(W, f)$ be a congruence pair of $G$ and $\wp(W, f) = \varrho$. We shall prove that $\varrho$ is a special congruence and $T(\varrho) = (W, f)$.
\bigskip

Let $S$ be a semigroup, let $\mathbf{T}$ be a relation on $S$ and denote the congruence generated by $\mathbf{T}$ as $\mathbf{T}^\sharp$. If $c, d$ in $S$ are such that $c = xay, d = xby$ for some $x, y \in S^1$, where either $(a, b) \in \mathbf{T}$ or $(b, a) \in \mathbf{T}$, we say that $c$ is connected to $d$ by an {\it elementary $\mathbf{T}$-transition} and write $c \rightarrow d$.

\begin{rem} \label{Trans}
Recall that the relation $\mathbf{R}$ consists of all pairs $(e_v e_v^\ast, v)$ for $v \in W$ and $(c^{f(c)}, s(c))$ for $c \in \cnw$ with $f(c) \in \mathbb{Z}^+$ and that $\wp(W, f) = \varrho = \mathbf{R}^\sharp$. The definition of an elementary $\mathbf{R}$-transition $c \rightarrow d$ in $\In$ can be adjusted as $c = xay, d = xby$ for some $x, y \in \In$ (not adjoined with $1$) where either $(a, b) \in \mathbf{R}$ or $(b, a) \in \mathbf(R)$. In fact, if either $x$ or $y$ equals to $1$, then either $x$ or $y$ can be replaced by some vertex since we have (GI1--3).
\end{rem}

\begin{lem} \label{Transition}
Let $\mathbf{T}$ be a relation on a semigroup $S$ and $a, b \in S$. Then $(a, b) \in \mathbf{T}^\sharp$ if and only if either $a = b$ or for some $n \in \mathbb{Z}^+$, there exists a sequence
$$
a = z_1 \rightarrow z_2 \rightarrow \cdots \rightarrow z_n = b
$$
of elementary $\mathbf{T}$-transitions connecting $a$ to $b$.
\end{lem}

\begin{lem} \label{vinWsigma}
For any $v \in V$, every element in $v \varrho$ is of the form either $\al \al^\ast$ or $\al c^{kf(c)} \al^\ast$, where $\al \in \pg$, $s(\al) = v$, $v(\al) \subseteq W$, and in the second case, $k \in \mathbb{Z}$, $r(\al) = s(c)$, $c \in \cnw$ with $f(c) \in \mathbb{Z}^+$.
\end{lem}

\begin{prf}
If $w \in v \varrho$, then we know from Lemma~\ref{Transition} that there exists a sequence
$$
v = z_1 \rightarrow z_2 \rightarrow \cdots \rightarrow z_n = w.
$$
of elementary $\mathbf{R}$-transitions. We now prove by induction on $n$. Obviously, the result is true if $n = 1$. Suppose that $z_{n-1}$ has one of the two forms in the lemma and that $z_{n-1} = \al \be^\ast x \xi \eta^\ast, z_n = \al \be^\ast y \xi \eta^\ast$ for some $\al \be^\ast, \xi \eta^\ast \in \In \setminus \{0\}$, where either $(x, y) \in \mathbf{R}$ or $(y, x) \in \mathbf{R}$. Then we have the following cases and subcases. \medskip \\
\textbf{Case 1.} $z_{n-1} = \ga \ga^\ast$ for some $\ga \in \pg$. Note $(\ref{multiplication})$ and that $v(\ga) \subseteq W$. \\
\textbf{Subcase 1.1.} $x \in W$ and $y = e_x e_x^\ast$. \\
\textbf{Subcase 1.1.1.} $\be = \xi \be_1$ for some $\be_1 \in \pg$. Then we get $z_{n-1} = \al \be_1^\ast \eta^\ast = \ga \ga^\ast$ which means that $\al = \ga = \eta \be_1$. Thus, if $|\be| = 0$, then we see that $z_n = \ga e_x e_x^\ast \ga^\ast$. Clearly, $v(\ga e_x) \subseteq W$. If $|\be| > 0$, then we know that $z_n = \al \be^\ast e_x e_x^\ast \xi \eta^\ast = \al \be^\ast \xi \eta^\ast = z_{n-1}$.  \\
\textbf{Subcase 1.1.2.} $\xi = \be \xi_1$ for some $\xi_1 \in \pg$. Then we get $z_{n-1} = \al \xi_1 \eta^\ast = \ga \ga^\ast$ which leads to $\al \xi_1 = \ga = \eta$. One can similarly obtain that either $z_n = \ga e_x e_x^\ast \ga^\ast$ or $z_n = z_{n-1}$ via discussing the two possibilities $|\xi| = 0$ and $|\xi| > 0$. \\
\textbf{Subcase 1.2.} $x \in W$ and $y = c^{f(c)}$ for some $c \in \cnw$ with $s(c) = x$ and $f(c) \in \mathbb{Z}^+$.  \\
\textbf{Subcase 1.2.1.} $\be = \xi \be_1$ for some $\be_1 \in \pg$. As in Subcase~1.1.1, we get $\al = \ga = \eta \be_1$. It follows from Lemma~\ref{Circle}(1) that $z_n = \al \be_1^\ast \xi^\ast c^{f(c)} \xi \eta^\ast = \al \be_1^\ast c_1^{f(c_1)} \eta^\ast$ for some $c_1 \in \cir c$. By Lemma~\ref{CircleAdd}, there exist $\be_2, \be_3 \in \pg$ and $l \in \mathbb{N}$ such that $\be_1 = c_1^l \be_2$ and $c_1 = \be_2 \be_3$. So we get $z_n = \eta c_1^l \be_2 \be_2^\ast (c_1^\ast)^l c_1^{f(c_1)} \eta^\ast$. Take $c_2 = \be_3 \be_2$. Clearly, $c_2 \in \cir c$. If $l \geq f(c_1) = f(c_2)$, then we have
$$
z_n = \eta c_1^l \be_2 \be_2^\ast (c_1^\ast)^{l - f(c_1)} \eta^\ast = \eta \be_2 c_2^l (c_2^\ast)^{l - f(c_2)} \be_2^\ast \eta^\ast = (\eta \be_2 c_2^{l - f(c_2)}) c_2^{f(c_2)} ((c_2^\ast)^{l - f(c_2)} \be_2^\ast \eta^\ast).
$$
Furthermore, $v(\eta \be_2 c_2^{l - f(c_2)}) \subseteq W$. If $l < f(c_1)$, then we have
$$
z_n = \eta c_1^l \be_2 \be_2^\ast c_1^{f(c_1) - l} \eta^\ast = \eta c_1^l \be_2 \be_2^\ast \be_2 \be_3 (c_1)^{f(c_1) - l - 1} \eta^\ast = \eta c_1^{f(c_1)} \eta^\ast.
$$
Obviously, $v(\eta) \subseteq W$. \\
\textbf{Subcase 1.2.2.} $\xi = \be \xi_1$ for some $\xi_1 \in \pg$. As in Case~1.1.2, we also get $\al \xi_1 = \ga = \eta$. It follows from Lemma~\ref{Circle}(1) that $z_n = \al \be^\ast c^{f(c)} \be \xi_1 \eta^\ast = \al c_1^{f(c_1)} \xi_1 \eta^\ast$ for some $c_1 \in \cir c$. So we see from Lemma~\ref{Circle}(2) that
$$
z_n = \al c_1^{f(c_1)} \xi_1 \xi_1^\ast \al^\ast = \al \xi_1 c_2^{f(c_2)} \xi_1^\ast \al^\ast
$$
for some $c_2 \in \cir c$. We already have $v(\al \xi_1) \subseteq W$. \\
\textbf{Subcase 1.3.} $x = e_y e_y^\ast$ and $y \in W$. Then we get $z_{n-1} = \al \be^\ast e_y e_y^\ast \xi \eta^\ast$. If $|\be| \neq 0$ or $|\xi| \neq 0$, then we easily see that $z_{n-1} = \al \be^\ast \xi \eta^\ast = z_n$. So we only need to consider the case where $|\be| = |\xi| = 0$. In this case, we have $\al e_y e_y^\ast \eta^\ast = \ga \ga^\ast$. So we get $\al e_y = \ga = \eta e_y$ which means that $z_n = \al \al^\ast$ with $v(\al) \subseteq W$. \\
\textbf{Subcase 1.4.} $x = c^{f(c)}$ and $y \in W$ for some $c \in \cnw$ with $s(c) = y$ and $f(c) \in \mathbb{Z}^+$.  \\
\textbf{Subcase 1.4.1.} $\be = \xi \be_1$ for some $\be_1 \in \pg$. Then we get $z_{n-1} = \al \be_1^\ast \xi^\ast c^{f(c)} \xi \eta^\ast$ and $z_n = \al \be_1^\ast \eta^\ast$. It follows from Lemma~\ref{Circle}(1) that there exists $c_1 \in \cir c$ such that $z_{n-1} = \al \be_1^\ast c_1^{f(c_1)} \eta^\ast$. By Lemma~\ref{CircleAdd}, we see that $\be_1 = c_1^l \be_2$ where $l \in \mathbb{N}$ and $c_1 = \be_2 \be_3$ for some $\be_3 \in \pg$. Clearly, $c_2 = \be_3 \be_2 \in \cir c$. If $l \geq f(c)$, then we obtain that $z_{n-1} = \al \be_2^\ast (c_1^\ast)^{l - f(c)} \eta^\ast = \ga \ga^\ast$ which leads to $\al = \ga = \eta c_1^{l - f(c)} \be_2$. Hence, we have
$$
z_n = \eta c_1^{l - f(c_1)} \be_2 \be_2^\ast (c_1^\ast)^l \eta^\ast = \eta \be_2 c_2^{l - f(c_2)} (c_2^\ast)^l \be_2^\ast \eta^\ast = \ga (c_2^\ast)^{f(c_2)} \ga^\ast
$$
as required. If $l < f(c_1)$, then we see that $z_{n-1} = \al \be_2^\ast c_1^{f(c_1) - l} \eta^\ast = \al \be_3 c_1^{f(c_1) - l -1} \eta^\ast = \ga \ga^\ast$ which leads to $\al \be_3 c_1^{f(c_1) - l -1} = \ga = \eta$. Hence, we have
$$
z_n = \al \be_2^\ast (c_1^\ast)^l (c_1^\ast)^{f(c_1) - l - 1} \be_3^\ast \al^\ast = \al (c_2^\ast)^{f(c_2)} \al^\ast.
$$
Obviously, $v(\al) \subseteq W$.  \\
\textbf{Subcase 1.4.2.} $\xi = \be \xi_1$ for some $\xi_1 \in \pg$. Then we get $z_{n-1} = \al \be^\ast c^{f(c)} \be \xi_1 \eta^\ast$ and $z_n = \al \xi_1 \eta^\ast$. It follows from Lemma~\ref{Circle}(1) that there exists $c_1 \in \cir c$ such that $z_{n-1} = \al c_1^{f(c_1)} \xi_1 \eta^\ast$. So we see that $\al c_1^{f(c)} \xi_1 = \ga = \eta$. Therefore we obtain from Lemma~\ref{Circle}(2) that
$$
z_n = \al \xi_1 \xi_1^\ast (c_1^\ast)^{f(c_1)} \al^\ast = \al (c_1^{f(c_1)} \xi_1 \xi_1^\ast)^\ast \al^\ast = \al \xi_1 (c_2^\ast)^{f(c_2)} \xi_1^\ast \al^\ast
$$
for some $c_2 \in \cir c$. Moreover, $v(\al \xi_1) \subseteq W$. \medskip \\
\textbf{Case 2.} $z_{n-1} = \ga c^{kf(c)} \ga^\ast$, where $\ga \in \pg$, $s(\ga) = v$, $v(\ga) \subseteq W$, $k \in \mathbb{Z}^+$, $r(\ga) = s(c)$ and $c \in \cnw$ with $f(c) \in \mathbb{Z}^+$.
\\
\textbf{Subcase 2.1.} $x \in W$ and $y = e_x e_x^\ast$. \\
\textbf{Subcase 2.1.1.} $\be = \xi \be_1$ for some $\be_1 \in \pg$. Then we get $z_{n-1} = \al \be_1^\ast \eta^\ast = \ga c^{kf(c)} \ga^\ast$ which means that $\al = \ga c^{kf(c)}$ and $\ga = \eta \be_1$. Thus, if $|\be| = 0$, then we see from Lemma~\ref{Circle}(2) that $z_n = \ga c^{kf(c)} e_x e_x^\ast \ga^\ast = \ga e_x c_1^{kf(c_1)} e_x^\ast \ga^\ast$ for some $c_1 \in \cir c$. Obviously, $v(\ga e_x) \subseteq W$. If $|\be| > 0$, then we know that $z_n = \al \be^\ast e_x e_x^\ast \xi \eta^\ast = \al \be^\ast \xi \eta^\ast = z_{n-1}$.  \\
\textbf{Subcase 2.1.2.} $\xi = \be \xi_1$ for some $\xi_1 \in \pg$. Then we get $z_{n-1} = \al \xi_1 \eta^\ast = \ga c^{kf(c)} \ga^\ast$ and similarly obtain that either $z_n = \ga e_x c_1^{kf(c_1)} e_x^\ast \ga^\ast$ for some $c_1 \in \cir c$ or $z_n = z_{n-1}$ via discuss the two possibilities $|\xi| = 0$ and $|\xi| > 0$. \\
\textbf{Subcase 2.2.} $x \in W$ and $y = d^{f(d)}$ for some $d \in \cnw$ with $s(d) = x$ and $f(d) \in \mathbb{Z}^+$. It is not difficult to see that $c$ and $d$ are cyclic permutations of each other since they are no-exit.  \\
\textbf{Subcase 2.2.1.} $\be = \xi \be_1$ for some $\be_1 \in \pg$. As in Subcase~2.1.1, we get $\al = \ga c^{kf(c)}$ and $\ga = \eta \be_1$. It follows from Lemma~\ref{Circle}(1) that $z_n = \al \be_1^\ast \xi^\ast d^{f(d)} \xi \eta^\ast = \al \be_1^\ast c_1^{f(c_1)} \eta^\ast$ for some $c_1 \in \cir c$. By Lemma~\ref{CircleAdd}, there exist $\be_2, \be_3 \in \pg$ and $l \in \mathbb{N}$ such that $\be_1 = c_1^l \be_2$ and $c_1 = \be_2 \be_3$. Moreover, we get $z_n = \ga c^{kf(c)} \be_2^\ast (c_1^\ast)^l c_1^{f(c_1)} \eta^\ast$. So we see that $c = \be_3 \be_2$ since $r(\be_2) = s(c)$. Hence, we see that
$$
z_n = \eta c_1^l \be_2 c^{kf(c)} \be_2^\ast (c_1^\ast)^l c_1^{f(c_1)} \eta^\ast = \eta \be_2 c^{kf(c)+l} \be_2^\ast (c_1^\ast)^l c_1^{f(c_1)} \eta^\ast.
$$
If $l \geq f(c)$, then we have
\begin{align*}
z_n & = \eta \be_2 c^{kf(c)+l} \be_2^\ast (c_1^\ast)^{l-f(c)} \eta^\ast = \eta \be_2 c^{kf(c)+l} (c^\ast)^{l-f(c)} \be_2^\ast \eta^\ast \\
& = (\eta \be_2 c^{l - f(c)}) c^{(k+1) f(c)} ((c^\ast)^{l - f(c)} \be_2^\ast \eta^\ast).
\end{align*}
We observe that $v(\eta \be_2 c_1^{l - f(c)}) \subseteq W$. If $l < f(c)$, then we have
$$
z_n = \eta c_1^{kf(c_1)+l} \be_2 \be_2^\ast c_1^{f(c_1)-l} \eta^\ast = \eta c_1^{kf(c_1)+l} \be_2 \be_2^\ast \be_2 \be_3 c_1^{f(c_1) - l - 1} \eta^\ast = \eta c_1^{(k+1)f(c_1)} \eta^\ast.
$$
Obviously, $v(\eta) \subseteq W$. \\
\textbf{Subcase 2.2.2.} $\xi = \be \xi_1$ for some $\xi_1 \in \pg$. As in Subcase~2.1.2, we also get $z_{n-1} = \al \xi_1 \eta^\ast = \ga c^{kf(c)} \ga^\ast$ so that $\al \xi_1 = \ga c^{kf(c)}$ and $\eta = \ga$. It follows from Lemma~\ref{Circle}(1) that $z_n = \al \be^\ast d^{f(d)} \be \xi_1 \eta^\ast = \al c_1^{f(c_1)} \xi_1 \eta^\ast$ for some $c_1 \in \cir c$. By Lemma~\ref{CircleAdd}, there exist $\xi_2, \xi_3 \in \pg$ and $l \in \mathbb{N}$ such that $\xi_1 = c_1^l \xi_2$ and $c_1 = \xi_2 \xi_3$. Let $c_2 = \xi_3 \xi_2$. Clearly, $c_2 \in \cir {c_1}$ and $\xi_1 = \xi_2 c_2^l$. It follows from $\al \xi_1 = \ga c^{kf(c)}$ that $c = c_2$. Thus we observe that
$$
z_n = \al c_1^{f(c_1)+l} \xi_2 \eta^\ast = \al \xi_2 c^{f(c)+l} \eta^\ast = \al \xi_1 c^{f(c)} \eta^\ast = \ga c^{(k+1)f(c)} \ga^\ast.
$$
We already have $v(\ga) \subseteq W$. \\
\textbf{Subcase 2.3.} $x = e_y e_y^\ast$ and $y \in W$. Then we get $z_{n-1} = \al \be^\ast e_y e_y^\ast \xi \eta^\ast$. If $|\be| \neq 0$ or $|\xi| \neq 0$, then we easily see that $z_{n-1} = \al \be^\ast \xi \eta^\ast = z_n$. So we only need to consider the case where $|\be| = |\xi| = 0$ which means that $\al e_y e_y^\ast \eta^\ast = \ga c^{kf(c)} \ga^\ast$. So we get $\al e_y = \ga c^{kf(c)}$ and $\ga = \eta e_y$ which lead to $\al e_y = \eta e_y c^{kf(c)} = \eta c_1^{kf(c_1)} e_y$. Hence we get $\al = \eta c_1^{kf(c)}$ so that $z_n = \al \eta^\ast = \eta c_1^{kf(c_1)} \eta^\ast$ with $v(\eta) \subseteq W$. \\
\textbf{Subcase 2.4.} $x = d^{f(c)}$ and $y \in W$ for some $d \in \cnw$ with $s(d) = y$ and $f(d) \in \mathbb{Z}^+$. Similar to Subcase~2.2, we observe that $c$ and $d$ are cyclic permutations of each other.  \\
\textbf{Subcase 2.4.1.} $\be = \xi \be_1$ for some $\be_1 \in \pg$. Then we get $z_{n-1} = \al \be_1^\ast \xi^\ast d^{f(d)} \xi \eta^\ast$ and $z_n = \al \be_1^\ast \eta^\ast$. It follows from Lemma~\ref{Circle}(1) that there exists $c_1 \in \cir c$ such that $z_{n-1} = \al \be_1^\ast c_1^{f(c_1)} \eta^\ast$. By Lemma~\ref{CircleAdd}, we see that $\be_1 = c_1^l \be_2$ where $l \in \mathbb{N}$ and $c_1 = \be_2 \be_3$ for some $\be_3 \in \pg$. If $l \geq f(c_1)$, then we obtain that $z_{n-1} = \al \be_2^\ast (c_1^\ast)^{l - f(c_1)} \eta^\ast = \ga c^{kf(c)} \ga^\ast$ which leads to $\al = \ga c^{kf(c)}$ and $\ga = \eta c_1^{l - f(c_1)} \be_2$. These also imply that $c = \be_3 \be_2$. Hence, we have
$$
z_n = \eta c_1^{l - f(c_1)} \be_2 c^{kf(c)} \be_2^\ast (c_1^\ast)^l \eta^\ast = \eta \be_2 c^{l - f(c)} c^{kf(c)} (c^\ast)^l \be_2^\ast \eta^\ast = (\eta \be_2 c^l) c^{(k-1)f(c)} ((c^\ast)^l \be_2^\ast \eta^\ast)
$$
and $v(\eta \be_2 c_1^l) \subseteq W$. If $l < f(c_1)$, then we see that
$$
z_{n-1} = \al \be_2^\ast c_1^{f(c_1) - l} \eta^\ast = \al \be_3 c_1^{f(c_1) - l -1} \eta^\ast = \ga c^{kf(c)} \ga^\ast
$$
which leads to $\al \be_3 c_1^{f(c_1) - l -1} = \ga c^{kf(c)}$, $\ga = \eta$ and $c_1 = c$. So we get $\al \be_3 = \ga c^{(k-1)f(c) + l + 1}$ which means that $\al = \ga c^{(k-1)f(c) + l} \be_2$. Clearly, $c_2 = \be_3 \be_2 \in \cir c$. Hence, we have
$$
z_n = \ga c^{(k-1)f(c) + l} \be_2 \be_2^\ast (c^\ast)^l \ga^\ast = (\ga \be_2 c_2^l) c_2^{(k-1)f(c_2)} ((c_2^\ast)^l \be_2^\ast \ga^\ast)
$$
and $v(\ga \be_2 c_2^l) \subseteq W$.  \\
\textbf{Subcase 2.4.2.} $\xi = \be \xi_1$ for some $\xi_1 \in \pg$. Then we get $z_{n-1} = \al \be^\ast d^{f(d)} \be \xi_1 \eta^\ast$ and $z_n = \al \xi_1 \eta^\ast$. It follows from Lemma~\ref{Circle}(1) that there exists $c_1 \in \cir c$ such that $z_{n-1} = \al c_1^{f(c_1)} \xi_1 \eta^\ast = \ga c^{kf(c)} \ga^\ast$. So we see that $\al c_1^{f(c_1)} \xi_1 = \ga c^{kf(c)}$ and $\ga = \eta$. Noticing Lemma~\ref{CircleAdd}, one has $\al c_1^{f(c_1)} \xi_1 = \al \xi_1 c^{f(c)} = \ga c^{kf(c)}$ which leads to $\al \xi_1 = \ga c^{(k-1)f(c)}$. Therefore we obtain that $z_n = \ga c^{(k-1)f(c)} \ga^\ast$. \medskip \\
\textbf{Case 3.} $z_{n-1} = \ga (c^\ast)^{kf(c)} \ga^\ast$, where $\ga \in \pg$, $s(\ga) = v$, $v(\ga) \subseteq W$, $k \in \mathbb{Z}^+$, $r(\ga) = s(c)$ and $c \in \cnw$ with $f(c) \in \mathbb{Z}^+$. With a similar discussion as in Case~2, we can see that $z_n$ has the required form.
\end{prf}

\begin{cor} \label{Special}
The congruence $\varrho$ is special.
\end{cor}

\begin{rem}
The relation $\mathbf{R}$ used in the proof of Proposition~\ref{vinWsigma} can be chosen smaller. $\mathbf{R}$ only need to consist of all pairs $(e_v e_v^\ast, v)$ for $v \in W$ and $(c^{f(c)}, s(c))$ for $c \in C_0$ such that $f(c) \in \mathbb{Z}^+$ and $|C_0 \cap \cir d| = 1$ for every $d \in \cnw$. This can be see from (CK1).
\end{rem}

\begin{prop} \label{PCP}
$T(\varrho) = (W, f)$.
\end{prop}

\begin{prf}
Let $T(\varrho) = (W_1, f_1)$. It is not difficult to see that $W \subseteq W_1$. Conversely, if $v \in W_1$, then we certainly have $(v, e_v e_v^\ast) \in \varrho$. It follows from Lemma~\ref{vinWsigma} and the uniqueness of the expression of every nonzero element in $\In$ that $v \in W$. So we get $W_1 = W$.

For $f$ and $f_1$, arbitrarily taking $c \in \cnw$ with $f(c) \in \mathbb{Z}^+$, we observe from the definition of $\varrho$ and Lemma~\ref{Divisible} that $f_1(c) \mid f(c)$ since $(c^{f(c)}, s(c)) \in \varrho$. On the other hand, let $f_1(c) \in \mathbb{Z}^+$. Again noticing Lemma~\ref{vinWsigma} and uniqueness of the expression of every nonzero element in $\In$, we obtain that there must exist $k \in \mathbb{Z}^+$ such that $c^{f_1(c)} = c^{kf(c)}$ since $(s(c), c^{f_1(c)}) \in \varrho$. Thus, we get $f(c) \mid f_1(c)$. In conclusion, we have $f_1 = f$.
\end{prf}

\medskip
Up to now, we have completed the proof of Theorem~\ref{Main1}. To end the section, we prove Corollary~\ref{Noetherian}. \medskip

Let $(H, W, f)$ be a congruence triple. Denote the set $\{v \in V \setminus H : |s^{-1}_{G \setminus H} (v)| = 1\}$ as $\bar{V}_H$. Obviously, $W \subseteq \bar{V}_H$. Define a relation $\leq$ on the set of all congruence triples of $G$: $(H_1, W_1, f_1) \leq (H_2, W_2, f_2)$ if $H_1 \subseteq H_2$, $W_1 \setminus H_2 \subseteq W_2$ and $f_2(c) \mid f_1(c)$ for any $c \in \cnh {W_1} \cap \cnh {W_2}$. We appoint that all positive integers and $\infty$ divide $\infty$.

\begin{lem} \label{PartialOrder}
The above relation $\leq$ is a partial order on the set of all congruence triples of $G$.
\end{lem}

\begin{prf}
Obviously, the relation is reflexive and antisymmetric. To see it is transitive, we suppose that $(H_1, W_1, f_1) \leq (H_2, W_2, f_2)$ and $(H_2, W_2, f_2) \leq (H_3, W_3, f_3)$. Then we certainly have $H_1 \subseteq H_2$, $H_2 \subseteq H_3$, $W_1 \setminus H_2 \subseteq W_2$, $W_2 \setminus H_3 \subseteq W_3$, $f_2(c_1) \mid f_1(c_1)$ for any $c_1 \in \cnh {W_1} \cap \cnh {W_2}$ and $f_3(c_2) \mid f_2(c_2)$ for any $c_2 \in \cnh {W_2} \cap \cnh {W_3}$. It follows that $W_1 \setminus H_3 = (W_1 \setminus H_2) \setminus H_3 \subseteq W_2 \setminus H_3 \subseteq W_3$. If $d \in \cnh {W_1} \cap \cnh {W_3}$, then we observe that $d \in \cnh {W_2}$ because $r(d) \not\in H_3$ implies $r(d) \not\in H_2$ and $r(e) \in H_1$ implies $r(e) \in H_2$ for any exit $e$ to $d$. So we have $f_3(d) \mid f_1(d)$ which leads to $(H_1, W_1, f_1) \leq (H_3, W_3, f_3)$.
\end{prf} \medskip

By Theorem~\ref{Main2}, the mapping $T$ is a one-to-one correspondence from the set of all congruences on $\In$ to the set of all congruence triples of $G$. Concretely, given a congruence $\rho$ on $\In$, the congruence triple $T(\rho) = (H, W, f)$ is defined as what follows.
\begin{align*}
H & = 0 \rho \cap V, \\
W & = \{ s(e) \in V : (e e^\ast, s(e)) \in \rho, r(e) \not\in H\},
\end{align*}
and for any cycle $c$ such that $v(c) \subseteq W$ and every exit $e$ to $c$ satisfies $r(e) \in H$,
$$
f(c) = \left\{ \begin{array}{ll}
\infty & \mbox{if for any }k \in \mathbb{Z}^+, (c^k, s(c)) \not\in \rho, \\
m & \mbox{if } m = \min\{n \in \mathbb{Z}^+ : (c^n, s(c)) \in \rho \}.
\end{array} \right.
$$
Arbitrarily pick congruences $\rho_1, \rho_2$ on $\In$ such that $\rho_1 \subseteq \rho_2$, $T(\rho_1) = (H_1, W_1, f_1)$ and $T(\rho_2) = (H_2, W_2, f_2)$. Notice Lemma~\ref{Divisible}. It is not difficult to see that $(H_1, W_1, f_1) \leq (H_2, W_2, f_2)$. That is to say, $T$ is order preserving. Hence, if $G$ is finite, then $\In$ must be congruence Noetherian.

\section{Some other applications}

Let $S$ be a semigroup. $S$ is called {\it congruence free} if the identity relation and the universal relation are the only congruences on $S$. If $S$ has a zero element, $S$ is called {\it 0-simple} if $S$ has only $\{0\}$ and $S$ as its ideals. Let $G = (V, E, r, s)$ be a graph. A pair of vertices $u$ and $v$ in $G$ is {\it strongly connected} if there exist a path from $u$ to $v$ and a path from $v$ to $u$. A graph is {\it strongly connected} if each pair of vertices is strongly connected. Notice the correspondence of the set of ideals in $\In$ and the set of hereditary subsets of $V$. The next corollary is obvious.

\begin{cor} \label{ZeroSimple}
The following are equivalent on $\In$:
\begin{description}
\item[$(1)$] $\In$ is 0-simple;
\item[$(2)$] the only hereditary subsets of $V$ are $\emptyset$ and $V$;
\item[$(3)$] $G$ is a strongly connected graph.
\end{description}
\end{cor}

The equivalence of (1) and (3) in the following corollary is proved in \cite[Theorem~10]{MesyanMitchell}.

\begin{cor} \label{ReesCon}
The following are equivalent:
\begin{description}
\item[$(1)$] every congruence on $\In$ is a Rees congruence;
\item[$(2)$] for every hereditary subset $H$ of $V$, $\bar{V}_H = \emptyset$;
\item[$(3)$] for every $e \in E$, there exists $\al \in \pg$ such that $e$ is an exit to $\al$, $s(e) = s(\al)$ and $r(e) = r(\al)$.
\end{description}
\end{cor}

\begin{prf}
Again noticing the correspondence of the set of ideals in $\In$ and the set of hereditary subsets of $V$, we see from the definition of congruence triples, Theorem~\ref{Main2} and its proof that (1) and (2) are equivalent.

Assume that there exists some $e_0 \in E$ such that either $e_0$ is not an exit to any path, or any path $\al$ to which $e_0$ is an exit and satisfies $s(e_0) = s(\al)$ does not end at $r(e_0)$. Taking $H_0 = \{ r(\al) : \al \in \pg \setminus (V \cup \{e_0\}), s(e_0) = s(\al)\}$, we see that $H_0$ is a hereditary subset of $V$ (probably empty). However, $\bar{V}_{H_0} \neq \emptyset$ since $s(e_0) \in \bar{V}_{H_0}$. Hence, we obtain that (2) implies (3). By the definition of hereditary subsets and $\bar{V}_H$, one easily sees that (3) implies (2).
\end{prf} \medskip

The equivalence of (1) and (3) in the next corollary is proved in \cite[Theorem~3]{AshHall} for graphs where there exists at most one edge from a vertex to another and in \cite[Corollary~11]{MesyanMitchell} for general cases.

\begin{cor}
The following are equivalent:
\begin{description}
\item[$(1)$] the semigroup $\In$ is congruence free;
\item[$(2)$] $\emptyset$ and $V$ are the only hereditary subsets and $\bar{V} = \emptyset$;
\item[$(3)$] the graph $G$ is strongly connected and no vertex of $G$ has index one.
\end{description}
\end{cor}

\begin{prf}
It follows from Theorem~\ref{Main2} and Corollary~\ref{ReesCon} that (1) and (2) are equivalent. Moreover, noticing that no vertex of $G$ has index one if and only if $\bar{V} = \emptyset$, we see that (2) is equivalent to (3) from Corollary~\ref{ZeroSimple}.
\end{prf} \medskip

\noindent {\bf Acknowledgements}

I would like to express our profound gratitude to Professor Efim Zelmanov for his invaluable guidance and help.


text:
\section{Introduction}
Hybrid mesons are quark-antiquark pairs having valence gluons as a structural component.
In some cases their quantum numbers are not accessible with quark models, and they
are therefore called exotic. Examples of these exotics are the $J^{PC}=0^{+-},1^{-+},\:\mbox{and}\;\,2^{+-}$ mesons.
Because these mesons contain valence gluons
their verification is one of the signature tests of Quantum Chromodynamics (QCD).

Efforts to determine properties of these hybrid exotic states
are unsettled from both experimental and theoretical viewpoints~\cite{Barnes:2003vy,Barnes:2000vn,Barnes:2000ns}.
The experimental efforts date back over a
decade~\cite{Alde:1988bv,Thompson:1997bs,Abele:1998gn,Adams:1998ff,Chung:1999we,Ivanov:2001rv,Dzierba2005},
and currently, considerable resources are being devoted to their future study.
The Jefferson Lab GlueX experimental program is designed to investigate exotic states.
Hybrid meson studies are also part of the COMPASS experiment at CERN
and the CLEOc program at Cornell.

Attempts to calculate decay widths of hybrid mesons have been made using the bag model \cite{Barnes:1982tx},
the quark model extended by gluon flux tube degrees of freedom \cite{Barnes:1995hc,Close:1995hc}
and in lattice QCD \cite{McNeile:2006bz,McNeile:2002az}.

Calculating properties of resonances using Euclidean lattice QCD simulations is
not straightforward for a variety of reasons.
On a finite lattice all states are bound, and furthermore, the lattice total energy of a
two-hadron state in a decay channel is typically larger than the energy of the
original hadron thus preventing decay.
Aspects of these points have been discussed by Michael~\cite{Michael:1989mf,Michael:2005},
DeGrand~\cite{Degrand:1991dg}, L{\"u}scher~\cite{Luscher:1991cf}, and by Lellouch
and L{\"u}scher~\cite{Lellouch:2000pv}.

Thus, lattice work on hybrid mesons has concentrated mainly on their mass spectrum. It can be
roughly classified in terms of heavy quark systems using static quarks with (excited) glue treated
in the Born-Oppenheimer approximation~\cite{Juge:1999ie,Juge:1998nc,Perantonis:1990dy}, using
non-relativistic QCD~\cite{Burch:2001tr,Drummond:1999db,Manke:1998tb}, and studies using actions
with both quenched and unquenched quark
dynamics~\cite{McNeile:1998cp,Bernard:1997ib,Bernard:1997bg,Lacock:1998be,Lacock:1998an,Lacock:1997ny,Lacock:1996vy,Bernard:2003cw}.
Lattice work on hadron resonances, of which there is very little at this time,
has recently been reviewed by Michael \cite{Michael:2005}.

Hybrid meson decay widths have been studied on the lattice in the heavy
quark limit~\cite{McNeile:2002az} and, recently, for light quarks \cite{McNeile:2006bz}.
In those studies the time dependence (slope) of a normalized transition matrix element
computed on the lattice is related to a decay width via Fermi's golden rule \cite{McNeile:2002fh}.
For this approach to work the lattice parameters have to be such that the resonance mass
comes out close to the threshhold of the decay channel.

We here choose to extract decay widths using L{\"u}scher's finite box
method~\cite{Luscher:1991cf,Luscher:1991ux,Cook:2006mc}. In principle L{\"u}scher's method is
rigorous:
The two-particle energy spectrum in a finite periodic box is related to continuum
elastic scattering amplitudes. The spectra allow calculation
of the scattering phase shifts at a discrete set of momenta owing to exact formulae derived by
L{\"u}scher. A decay width can then be extracted by fitting a Breit-Wigner function,
provided a resonant state is actually present.
The applicability of this method to extract scattering phase shifts has been demonstrated
for the O(3) non-linear sigma model in 1+1
dimensions~\cite{Luscher:1990ck}, the O(4) non-linear sigma model in 3+1
dimensions~\cite{Gockeler:1994rx,Zimmermann:1993kx}, meson-meson
scattering in 2+1 dimensions using QED~\cite{Fiebig:1994qi}, and resonance scattering of two coupled
Ising systems~\cite{Gat93b,Gat93a}.

Application of L{\"u}scher's method to our desired goal requires a set of operators that couple to
the hybrid meson state and to appropriate two-meson systems matching a decay channel. The exotic
$1^{-+}$ meson can decay into $\pi b_1$, $\pi f_1$, and $\pi a_1$, which are relative S-wave
channels. Other decays are possible, but those involve relative P-waves, where the relative momentum is
at least $2\pi/L$, and thus give rise to large upward energy shifts which makes the
simulation more difficult \cite{Degrand:1991dg,McNeile:2002fh,Loft:1988sy}.
We note that experimentally \cite{PDBook:2004} the three mesons $b_1(1235)$, $a_1(1260)$, $f_1(1285)$
are close in mass, and that the $a_1$ is a vector meson with $C=+$ so that it combines
naturally with $C=0$ of the (neutral) pion to the required charge conjugation, $C=+$, of the
exotic meson.
Therefore, in this study, we model our two-hadron operator after the $\pi a_1$ decay channel.

\section{\label{sec:lat}Lattice Parameters}
Obtaining excited state spectra from correlation matrices that involve
two-hadron operators presents a numerical challenge in itself.
On top of this more analysis steps are required to obtain a decay width,
which essentially amounts to extracting a ``derivative'' quantity from the simulation.
While statistical errors, and also discretization errors to some extent, can be easily
controlled this is not true for the systematic uncertainties indigenous to a task of
this kind. For this reason we have refrained from using large computing
resources and therefore employed a very simple lattice action and moderate lattice
sizes and pion masses.

Simulations were performed using the Wilson gauge field action and Wilson
fermions in quenched approximation on anisotropic lattices.
We will present results
from 200 gauge configurations on $12^3\times24$ and $10^3\times24$ lattices.

The definition of the coupling parameters used for the Wilson gauge action
\begin{equation}\label{puregluon}
S_{g}[U]= \sum_{x}\sum_{1\le \mu < \nu \le 4}\beta_{\mu\nu}\biggl(1-\frac{1}{3}\mbox{\,Re\,Tr\,}(U_{\mu\nu}(x))\biggr)
\end{equation}
is given by
\begin{equation}
\beta_{\mu\nu}=\beta \frac{a_1 a_2 a_3 a_4}{(a_\mu a_\nu)^2}\,,
\end{equation}
where $\mu\nu$ denotes plaquette planes and $a_1=a_2=a_3=:a_s$ is the spatial
and $a_4=:a_t$ the temporal lattice constant. We have chosen the (bare) anisotropy
$\xi=a_s/a_t=2$ and $\beta=6.15$ for the global coupling parameter.
Parameters for the anisotropic fermion matrix
\begin{eqnarray}\label{quarkm}
Q(x,y)=\mathbf{1}\,\delta_{x,y}-\sum_{\mu}\kappa_{\mu}\biggl((1-\gamma_{\mu})U_{\mu}(x)\delta_{x+\hat{\mu},y} &&\nonumber \\
+(1+\gamma_\mu)U^{\dagger}_{\mu}(y)\delta_{x,y+\hat{\mu}}\biggr) &&
\end{eqnarray}
are given by
\begin{equation}\label{kappamu}
\kappa_{\mu}=\kappa\frac{4}{a_{\mu}\sum_{\lambda}\frac {1}{a_\lambda}}
\end{equation}
where $\kappa$ is a global hopping parameter.
With these conventions the relation of the latter to the (bare) Wilson quark mass
parameter $m_q$ is
\begin{equation}
\kappa=\frac{\sum_{\lambda}\frac{1}{a_{\lambda}}}{8(m_q+\sum_{\lambda}\frac{1}{a_{\lambda}})}\,,
\end{equation}
which identifies $\kappa_c=0.125$ as the critical value.

Correlation functions for mesons from standard local operators $\pi\sim\gamma_5$,
$\rho\sim\gamma_i$, and $a_1\sim\gamma_i\gamma_5$, $i=1,2,3$ were
constructed as a matter of course employing three iterations of quark field
smearing \cite{Alexandrou:1994ti} and gauge field fuzzing \cite{Alb87a}.
Also, within this setting, we adopted the hybrid meson operator proposed
in \cite{Bernard:1997ex} with magnetic type gluons
\begin{equation}\label{hyopert}
O_{\h}(t)= \sum_{1\le i < j \le 3}
\sum_{\vec{x}}\bar{d}_{a}(\vec{x}t) \gamma_{i} u_{b}(\vec{x}t) \,[F^{ab}_{ij}(\vec{x}t)- F^{\dagger ab}_{ij}(\vec{x}t)]\,,
\end{equation}
where $a,b$ denote color indices and $F_{ij}(x)$ is a product of SU(3) link matrices arranged in a
clover pattern
\begin{eqnarray}\label{gluonfields}
\lefteqn{F_{\mu\nu}(x) = U_{\mu}(x)U_{\nu}(x+\hat{\mu})U^{\dagger}_{\mu}(x+\hat{\nu})U^{\dagger}_{\nu}(x)\notag}\\
&+& U_{\nu}(x)U^{\dagger}_{\mu}(x-\hat{\mu}+\hat{\nu})U^{\dagger}_{\nu}(x-\hat{\mu})U_{\mu}(x-\hat{\mu})\notag\\
&+& U^{\dagger}_{\mu}(x-\hat{\mu})U^{\dagger}_{\nu}(x-\hat{\mu}-\hat{\nu})U_{\mu}(x-\hat{\mu}-\hat{\nu})
U_{\nu}(x-\hat{\nu})\notag\\
&+& U^{\dagger}_{\nu}(x-\hat{\nu})U_{\mu}(x-\hat{\nu})U_{\nu}(x+\hat{\mu}-\hat{\nu})U^{\dagger}_{\mu}(x)\,,
\end{eqnarray}
which is used in the spatial planes only. Under parity we have
${\cal P}O_{h}(t){\cal P}^{-1} = -O_{h}(t)$,
while for the charge neutral ($\bar{d}u\rightarrow \bar{u}u,\bar{d}d$ ) version $O_{h^{0}}(t)$ of
(\ref{hyopert}) under charge conjugation the derivation of
${\cal C}O_{h^{0}}(t){\cal C}^{-1} = -O_{h^{0}}(t)$
relies on the presence of $F-F^\dagger$, specifically ${\cal C}F_{ij}(x){\cal C}^{-1}=F^\ast_{ij}(x)$.
We have enforced this relation in the simulation:
Observing that $S_g[U]=S_g[U^\ast]$ the configurations
$[U]$ and $[U^\ast]$ are equally probable. Thus with each $[U]$ in the ensemble of 200
configurations we also include $[U^\ast]$ and compute fermion propagators for both of those.
This strategy doubles the number of fermion propagators that need to be computed, however,
charge conjugation is now numerically exact, and this also appears to be the reason for an observed
noise reduction of simulation signals.

Meson masses were obtained at four values of the hopping parameter $\kappa$,
see Table~\ref{tab:hopping}.
A multiple mass inverter \cite{Frommer:1995ik} was used to compute propagators.
The resulting ground state masses, coming from three smearing iterations, sources set at $t=3$,
and effective mass function fits in the range $t=6\ldots 11$, are also
listed in Tab.~\ref{tab:hopping}.
\begin{table}[h]
\caption{\label{tab:hopping}List of hopping parameters $\kappa$ and the resulting
$\pi$, $\rho$, $a_1$ and $h$ meson masses in units of the temporal lattice constant $a_t$
for lattices $12^{3}\times24$ (upper table) and $10^{3}\times24$ (lower table).}
\begin{ruledtabular}
\begin{tabular}{ccccc}
${\kappa}$ & ${a_{t}m_{\pi}}$ & ${a_{t}m_{\rho}}$ & $ {a_{t}m_{a_1}}$ & ${a_{t}m_{\h}}$ \\
\colrule
0.140 & 0.53(4) & 0.55(3) & 0.65(4) & 0.63(25) \\
0.136 & 0.64(3) & 0.65(3) & 0.75(3) & 0.75(21) \\
0.132 & 0.75(3) & 0.75(3) & 0.86(3) & 0.85(26) \\
0.128 & 0.85(4) & 0.85(3) & 0.96(3) & 0.95(24) \\
\colrule
0.140 & 0.54(5) & 0.54(3) & 0.64(4) & 0.62(24) \\
0.136 & 0.65(3) & 0.62(3) & 0.75(3) & 0.71(23) \\
0.132 & 0.74(3) & 0.73(3) & 0.85(4) & 0.81(25) \\
0.128 & 0.85(4) & 0.83(3) & 0.96(3) & 0.91(22) \\
\end{tabular}
\end{ruledtabular}
\end{table}

In order to allow extrapolations to the small pion mass region it is useful
to study the dependence of the computed $\rho$, $a_{1}$ and $\h$ masses on $M_\pi=a_{t}m_\pi$.
Predictions for this dependence may in principle come from chiral perturbation
theory \cite{BernsteinHolstein:1995}, and will depend on the baryon being studied.
For a baryon of mass $M=a_{t}m$ the expression
\begin{equation}\label{chiPT}
M \approx p_0+p_2\,M_{\pi}^2+p_3\,M_{\pi}^3+p_4\,M_{\pi}^4+q\,M_{\pi}^4\ln(M_\pi)
\end{equation}
contains a collection of terms typical for $\chi$PT inspired
models \cite{Smit:2002sm,Wright:2002wr}.
In the case of the hybrid exotic meson, for example, the authors of \cite{Hedditch:2005zf}
retain only the even polynomial in (\ref{chiPT}).
For the $\pi a_1$ decay channel, which is mostly relevant in this work,
no predictions for the dependence of the spectral masses, say $W$, on $M_\pi$ are
available. A three parameter model that reflects features of (\ref{chiPT}) is
\begin{equation}\label{fitln}
W=p+qx+r\ln (1+x) \quad\,\, \mbox{with} \,\,\, x=(a_{t}m_\pi)^2\,.
\end{equation}
The logarithmic term is purely heuristic. Its role is to provide curvature to the model,
just like the last three terms in (\ref{chiPT}) do while vanishing as $M_{\pi}\rightarrow 0$.
As it turns out this model yields fits that are on average optimal on our spectral
data for the combined $\h$ and $\pi a_1$ systems.
Replacing the logarithmic term in (\ref{fitln}) with $x^{3/2}$ yields nearly identical
results.

We consistently use (\ref{fitln}) to fit all masses emerging from the simulation.
Examples for the mesons listed in Tab.~\ref{tab:hopping} are shown in
Fig.~\ref{fig:effa1pionhyb}.

In particular the upper panel of Fig.~\ref{fig:effa1pionhyb} exhibits
a level crossing between the hybrid meson mass and the $\pi+a_1$ mass,
assuming a relative S-wave for the latter. The level crossing emerges near
$x\simeq 0$ which is only reached through extrapolation.
The lower panel illustrates the effect of P-wave~vs.~S-wave decay on the lattice.
The mass of the $\pi+\pi$ system is shown with pions having lattice
momenta $\pm 2\pi/(a_sL)$. Clearly a level crossing with the $\rho$ meson mass is
harder to achieve.
\begin{figure}[h]
\includegraphics[height=82mm,angle=90]{effect_a1pion_hybridmass_vs_pionsq_12_rev1.eps}
\includegraphics[height=82mm,angle=90]{effect_pionpion_rhomass_vs_pionsq_12_rev1.eps}
\caption{\label{fig:effa1pionhyb}Combinations of masses, as indicated, obtained from single
meson operators versus the squared pion mass $x=(a_tm_\pi)^2$ and fits with the model
(\protect\ref{fitln}).
The relative S-wave $\pi+a_1$ mass (upper panel) reveals a level crossing near
zero pion mass through extrapolation.
The $\pi+\pi$ mass is shown with pions having lattice
momenta $\pm 2\pi/(a_sL)$.
The extrapolation of the $\rho$ meson mass to $x=0$ is used to set
the physical scale.}
\end{figure}

The extrapolated $\rho$ meson mass, at $x=0$, shall be used to
set the physical mass or length scale for this simulation. We obtain
$a_t=0.33(5)\,\mbox{GeV}^{-1}\,=0.07(1)\,\mbox{fm}$ ($m_{\rho}=776\,\mbox{MeV}$).
If the $a_1$ meson is used instead the scale is
$a_t=0.30(3)\,\mbox{GeV}^{-1}\,=0.06(1)\,\mbox{fm}$ ($m_{a_1}=1230\,\mbox{MeV}$).
The above numbers are based on the $12^{3}\times 24$ lattice.
Unless otherwise indicated
we will quote results using the $\rho$ meson to set the scale.

As a sideline it is interesting to note that the level crossing seen in
Fig.~\ref{fig:effa1pionhyb}, using both of the above scales, thus occurs
within 1.35--1.49GeV, which overlaps with the experimental mass of the $\pi(1400)$
resonance, according to \cite{PDBook:2004}. Indeed the $\pi(1400)$ has the quantum
numbers $1^{-+}$ of the hybrid exotic meson.
This observation coincides with the findings of \cite{Hedditch:2005zf}.

\section{Correlation matrix}
The description of $h\rightarrow\pi+a_{1}$
requires an operator for the two-meson decay channel with suitable
quantum numbers. Consider
\begin{eqnarray}\label{piaone}
\lefteqn{O_{\pi^{+}a^{0}_{1};k,\vec{r}\,}(t)=\sum_{\vec{x}}\sum_{\vec{y}}\delta_{\vec{x}-\vec{y},\vec{r}}
\;\bar{d}_a(\vec{x}t)\gamma_{5}u_{a}(\vec{x}t)\notag}&&\\
&&[\bar{d}_b(\vec{y}t)\gamma_{5}\gamma_{k}d_{b}(\vec{y}t)
+\bar{u}_b(\vec{y}t)\gamma_{5}\gamma_{k}\,u_{b}(\vec{y}t)]
\end{eqnarray}
where $k=1,2,3$. The relative distance $\vec{r}$ may in principle be used to construct
operators that transform according to an irrep of the hypercubic group. However, the
simplest choice $\vec{r}=\vec{0}$ already leads to a viable operator.
Summing over all spatial directions we thus adopt
\begin{equation}\label{pia11}
O_{\pi a_1}(t)=\sum_{k=1}^{3}O_{\pi^{+}a^{0}_{1};k,\vec{0}\,}(t)
\end{equation}
for this simulation.

The operators (\ref{hyopert}) and (\ref{pia11}) are the basis for calculating
correlation functions
\begin{equation}\label{corXY}
C_{XY}(t,t_0)=
\langle O_{X}(t) O_{Y}^{\dagger}(t_0)\rangle -\langle O_{X}(t)\rangle\langle O_{Y}^{\dagger}(t_0)\rangle\,.
\end{equation}
Here $X$ and $Y$ stand for $h$ or $\pi a_1$, and thus establish a \mbox{$2\times2$} correlation matrix.
The separable terms in (\ref{corXY}) are zero because of the quark flavor assignment in $\h$ and
$\pi a_1$.
The remaining (non-separable) terms in (\ref{corXY}) contain contractions between quark fields at
equal times when worked out with Wick's theorem.
For example, showing flavor structure only,
\begin{equation}
C_{h,\pi a_1}(t,t_0) \sim\langle(\bar{d}u)_t (\bar{d}d\bar{u}d+\bar{u}u\bar{u}d)_{t_0}\rangle
\end{equation}
with time arguments $t$ and $t_0$ as indicated. Equal-time contractions
:$d\bar{d}$: and :$u\bar{u}$: occur only at the source time slice $t_0$.
The corresponding propagator elements $Q^{-1}(\vec{x}t_{0},\vec{y}t_0)$
are calculated by default. This is different for
\begin{equation}\label{Cpia1}
C_{\pi a_1,\pi a_1}(t,t_0) \sim\langle(\bar{d}u\bar{d}d+\bar{d}u\bar{u}u)_t
(\bar{d}d\bar{u}d+\bar{u}u\bar{u}d)_{t_0}\rangle
\end{equation}
where we encounter equal time contractions :$d\bar{d}$: and :$u\bar{u}$: at $t > t_0$.
The computation of $Q^{-1}(\vec{x}t,\vec{y}t)$ is very resource intensive and,
if stochastic estimation is used \cite{Michael:1998sg}, then it contributes additional noise.

We shall now argue that this problem can be circumvented:
At $m_d=m_u$ the contractions
:$d\bar{d}$: and :$u\bar{u}$: give rise to the same propagator elements $Q^{-1}(\vec{x}t,\vec{y}t)$.
Thus, replacing the $(\ldots)_t$ term in (\ref{Cpia1}) by $(2\bar{d}u\bar{d}d)_{t}$ and
reinstating the $\gamma$-matrices from (\ref{piaone})
we observe that $\bar{d}d\sim\bar{d}\gamma_{5}\gamma_{k}d$ couples to $a_1$ and $f_1$ mesons.
Their masses are close however, 1230MeV and 1282MeV respectively \cite{PDBook:2004}.
Invoking a similar argument, altering the quark flavor $d\rightarrow s$ in the above operator
entails
$\bar{d}\gamma_5\gamma_kd\longrightarrow\bar{d}\gamma_5\gamma_ks\sim K_{1}$
and, again, should not significantly alter the mass spectra because
the $K_1$ meson mass of 1270MeV \cite{PDBook:2004} again is close to that of the $a_1$ meson.
In terms of (\ref{Cpia1}) the effect is
\begin{equation}
C_{\pi a_1,\pi a_1}(t,t_0)\rightarrow
\langle(2\bar{d}u\bar{d}s)_{t}(2\bar{s}d\bar{u}d)_{t_0}\rangle\,,\notag
\end{equation}
which now has no equal time contractions, but otherwise is not different from
(\ref{Cpia1}) when worked out in terms of quark propagators.
Hence, dropping equal time contractions in (\ref{Cpia1})
should have little effect on the mass spectrum of the $\pi a_{1}$ system,
and ultimately, on the resulting decay width, because L{\"u}scher's method
for computing scattering phase shifts exclusively relies on the mass spectra, in a finite box.

We emphasize that the correlator element (\ref{Cpia1}) is worked out using the
quark flavor structure exactly as it emerges from (\ref{piaone}) except that equal
time contractions are neglected. All other matrix elements are not effected.

Finally, we do not explicitly compute $C_{\pi a_1,h}(t,t_0)$ but rather
infer it from the hermiticity of the correlation matrix.

\section{Analysis}

For every operator used in this simulation up to three iterations of quark field
smearing \cite{Alexandrou:1994ti} and gauge field fuzzing \cite{Alb87a}
were employed, using 2.5 as a strength parameter in both cases.
The $2\times2$ correlation matrix (\ref{corXY}) thus
expands to size $6\times6$,
\begin{equation}\label{calC}
{\mathcal C}(t,t_0)=\begin{pmatrix}
C_{\h\{ \},\h\{ \}}(t,t_0) & C_{\h\{ \},\pi a_1\{ \}}(t,t_0) \\
C_{\pi a_1\{ \},\h\{ \}}(t,t_0) & C_{\pi a_1\{ \},\pi a_1\{ \}}(t,t_0)\end{pmatrix}\,,
\end{equation}
where the entries are $3\times 3$ matrices with elements
$C_{X\{k\},Y\{\ell\}}(t,t_0)$ built from operators $O_{X\{k\}}(t,t_0)$, etc, with $k=1,2,3$ levels
of fuzzing and smearing.
The latter is done identically at both source and sink, and thus the matrix ${\mathcal C}(t,t_0)$
is hermitian by construction.

A standard analysis method
is based on solving the generalized eigenvalue problem \cite{Luscher:1990ck}
\begin{equation}\label{eigen}
{\mathcal C}(t,t_0)\Psi(t)={\mathcal C}(t_1,t_0)\Psi(t)\Lambda(t)
\end{equation}
where $t_1$ is fixed, $\Psi(t)$ is an $N\times N$ matrix, its columns being the generalized
eigenvectors, and $\Lambda(t)$ is real diagonal.
We further require that ${\mathcal C}(t_1,t_0)$ be positive
definite. To ensure the latter $t_1$ should be an ``early" time slice, here we use
$t_{1}-t_{0}=4$. The generalized eigenvalue problem (\ref{eigen}) can then be cast into
an ordinary one by first diagonalizing
\begin{equation}\label{ceigen}
{\mathcal C}(t_1,t_0) = V(t_1,t_0) D(t_1,t_0) V^{\dag}(t_1,t_0)\,,
\end{equation}
where $V(t_1,t_0)$ is unitary and $D(t_1,t_0)$ is real diagonal and positive definite.
Inserting (\ref{ceigen}) into (\ref{eigen}) we are lead to define
\begin{equation}\label{Chatei}
\widehat{C}(t)=\frac{1}{\sqrt{D(t_1,t_0)}}
V^{\dag}(t_1,t_0) {\mathcal C}(t,t_0) V(t_1,t_0) \frac{1}{\sqrt{D(t_1,t_0)}}\,.
\end{equation}
It's eigenvalues are the same as those of the generalized problem (\ref{eigen}),
$\Lambda(t)=\rm{diag}(\lambda_{1}(t)\ldots\lambda_{N}(t))$.
Constructing $\widehat{C}(t)$ merely amounts to
a linear transformation among the set of operators $O_{X\{k\}}(t,t_0)$ that define
the correlation matrix.
Because of $\widehat{C}(t_1)=\openone$, the transformed set of operators
create quantum states that are orthogonal and normalized at $t=t_1$. Provided that these
match the ``true" states of the theory, the eigenvectors of $\widehat{C}(t)$
will stay orthogonal as $t\ge t_1$ increases to the extent
allowed by the errors of the simulation. Consequently $\widehat{C}(t)$ will
be diagonal dominated for $t\ge t_1$.

From an analysis point of view there are now two options: One may diagonalize
$\widehat{C}(t)$ on each time slice $t$ separately and thus obtain the eigenvalues
$\lambda_{n}(t),\,n=1 \ldots N$, as proposed in \cite{Luscher:1990ck}.
Alternatively, the diagonal elements of $\widehat{C}(t)$ may
be taken as an approximation to its eigenvalues,
\begin{equation}\label{project}
\lambda_{n}(t)\approx \widehat{ C}_{nn}(t)\ ,\ n=1 \ldots N\,,
\end{equation}
which involves projecting $\widehat{C}(t)$ into the eigenspaces at fixed
time slice $t_1$, see (\ref{Chatei}).
The latter approach has the advantage that statistical fluctuations
are reduced, see Fig.~\ref{comparediag} for a comparison.
This is plausible because fluctuations of the eigenvector components are effectively frozen.
It also has the advantage to tag the eigenvalues to a specific eigenvector,
which is important for tracking the quark mass dependence of the spectral levels.
\begin{figure}
\includegraphics[height=82mm,angle=90]{correl_matrix_plot_10_diageveryslice_witherrors_rev1.eps}
\includegraphics[height=82mm,angle=90]{correl_matrix_plot_10_witherrors_rev1.eps}
\caption{\label{comparediag}Comparison of eigenvalues obtained from
diagonalizing $\widehat{C}(t)$ on every time slice
(upper panel) versus projecting to time slice $t=5$ (lower panel), with $t_0=1$.
Results are for the $10^3\times24$ lattice at the lightest pion mass. Fluctuations
are much reduced using the projection technique.}
\end{figure}

Effective mass function plateaus typically develop in the time interval
$5\lesssim t-t_0 \lesssim 10$, or so. In this region
the projection technique yields more stable results, particularly for the
excited states. An example for the $12^3 \times 24$ lattice at
$\kappa=0.140$ is shown in Fig.~\ref{efftwo}.
\begin{figure}
\includegraphics[height=82mm,angle=90]{effectmasses_lamda2and5_12.eps}
\caption{\label{efftwo}Example of effective mass functions for two (projected) eigenvalues
of the correlation matrix $\widehat{C}(t)$ on the
$12^3 \times 24$ lattice at the lightest pion mass. The higher mass comes from $\lambda_5(t)$ and
the lower mass from $\lambda_2(t)$.}
\end{figure}

Thus we will continue our analysis with the projected
correlators and, for simplicity, refer to $\widehat{C}_{nn}(t)$ as eigenvalues $\lambda_{n}(t)$.
Those then give rise to the spectral energies $W_n,n=1\dots 6$, listed in
Tab.~\ref{tab:enerspec2}.
\begin{table}[h]
\caption{\label{tab:enerspec2}Energy spectra $W_n, n=1\ldots 6$ from the $12^{3}\times24$
lattice (upper table) and the $10^{3}\times24$ lattice (lower table) at four pion masses.}
\begin{ruledtabular}
\begin{tabular} {lllll}
$\kappa$ & 0.140 & 0.136 & 0.132 & 0.128 \\
\colrule
$a_{t}W_{1}$ & 0.55(14) & 0.69(2)  & 0.74(4)  & 0.82(10) \\
$a_{t}W_{2}$ & 0.69(11) & 0.79(3)  & 0.89(4)  & 0.93(9)  \\
$a_{t}W_{3}$ & 0.81(10) & 0.81(4)  & 0.91(4)  & 1.03(7)  \\
$a_{t}W_{4}$ & 0.88(13) & 1.03(11) & 1.18(10) & 1.26(5)  \\
$a_{t}W_{5}$ & 1.21(9)  & 1.13(3)  & 1.25(3)  & 1.31(4)  \\
$a_{t}W_{6}$ & 0.96(5)  & 1.42(4)  & 1.62(3)  & 1.85(4)  \\
\colrule
$a_{t}W_{1}$ & 0.57(7)  & 0.69(11)  & 0.75(11) & 0.78(6)  \\
$a_{t}W_{2}$ & 0.59(11) & 0.71(5)   & 0.78(4)  & 0.81(5)  \\
$a_{t}W_{3}$ & 0.61(6)  & 0.74(5)   & 0.80(4)  & 0.91(5)  \\
$a_{t}W_{4}$ & 0.91(11) & 1.12(8)   & 1.14(13) & 1.24(9)  \\
$a_{t}W_{5}$ & 1.20(9)  & 1.42(11) & 1.64(9)  & 1.78(9)  \\
$a_{t}W_{6}$ & 1.31(6)  & 1.43(6)   & 1.55(9)  & 1.62(13) \\
\end{tabular}
\end{ruledtabular}
\end{table}

Fits to those spectra with the model (\ref{fitln}) are shown in Fig.~\ref{spectra}.
This figure sheds light on the volume dependence of the spectral levels.
Evidently the ground state mass is relatively stable against changing the lattice volume.
On the other hand the effect on excited states is clearly significant, even to the extent
that level crossing patterns differ for some of the states.
This should not be surprising because excited levels are likely to describe two-meson
states, which are spatially large.
Nevertheless, anticipating results, the volume effect on the scattering phase shift
ultimately turns out to be only moderate.
\begin{figure}
\includegraphics[height=82mm,angle=90]{effect_lamdaall_log_12_rev1.eps}
\includegraphics[height=82mm,angle=90]{effect_lamdaall_log_10_rev1.eps}
\caption{\label{spectra}Mass spectra obtained from the $12^3 \times 24$ and $10^3 \times 24$
lattices versus $x=(a_tm_\pi)^2$. Error bars are omitted for clarity except for one level where
the errors shown are typical of all levels.
Fits with the model (\protect\ref{fitln}) are shown as lines.}
\end{figure}

Another comment on Fig.~\ref{spectra} is that, although the $6\times 6$ correlation matrix
gives rise to six eigenvalues, the number of physical states
on the lattice
is likely to be a lesser number because, typically, hadronic level spacings are of the order of
a few hundred MeV. Thus we entertain the possibility that the group of the three lower levels
in Fig.~\ref{spectra} describe the same state, the ground state, whereas the upper levels
belong to two-meson states with some degree of interaction energy due to their relative motion.
This point will become more plausible in terms of the corresponding scattering phase shifts.

\section{Scattering phase shifts}

The spectral energies $W_n$ computed on the lattice give rise to a discrete set of relative
$\pi$+$a_1$ momenta $k_n$ by solving the relativistic dispersion relation
\begin{equation}\label{reldisper}
W_{n}=\sqrt{m_{\pi}^2+k^{2}_{n}}+\sqrt{m_{a_1}^2+k^{2}_{n}}\,.
\end{equation}
Note that the resulting momenta $k_n$ relate to spectral masses and thus are continuous
numbers (not subject to lattice discretization).
Only those levels $W_{n}$ will be used that fall within the elastic region,
\begin{equation}\label{welastic}
(m_{\pi}+m_{a_1})< W_{n} < 2 \, (m_{\pi}+m_{a_1})\,.
\end{equation}
Continuum S-wave scattering phase shifts $\delta_0(k)$ are then computed at a discrete
set of momenta $k_n$ using L{\"u}scher's formula \cite{Luscher:1991cf},
\begin{equation}
\tan\delta_0(k_n)=-\frac{\pi^{3/2}q_{n}}{{\cal Z}(1;q_{n}^2)} \quad , \quad q_{n}=
\frac{k_{n}L_{s}}{2\pi}\,.
\end{equation}
Here ${\cal Z}(1;q^2)$ is a generalized $\zeta$-function,
and $L_s=La_s$ is the physical size of the spatial box, using the bare anisotropy $a_s=2a_t$.

If the number of available data points is sufficient, then one may attempt a fit
to a Breit-Wigner function \cite{Tay72},
\begin{eqnarray}
\label{BWmodel}
\tan\delta_0(k)&=&\frac{\Gamma/2}{E_{0}-W(k)}\\
\label{BWmodel1}
\mbox{where} \quad W(k)&=&\sqrt{m_{\pi}^2+k^{2}}+\sqrt{m_{a_1}^2+k^{2}}\,.
\end{eqnarray}
The resonance energy $E_0$ and the decay width $\Gamma$ are fit parameters.
However, a successful fit can only be expected if the underlying physics indeed
supports an isolated resonance.
Such a fit actually fails for all spectra computed at the four pion masses,
or rather $x=(a_tm_\pi)^2$, as they appear in Fig.~\ref{spectra}.
This is not be surprising because those data points are far away from a level
crossing between the $\h$ and the $\pi$+$a_1$ masses as evident
from Fig.~\ref{fig:effa1pionhyb}.
It is necessary to extrapolate the spectral masses to $x=0$ near the level crossing.
The model (\ref{fitln}) has been used for this purpose.
We present the extrapolated spectra in Tab.~\ref{tab:eigenmoment1} along with
the corresponding momenta $k_n$ and scattering phase shifts $\delta_0(k_n)$
for those levels which fall into the elastic region (\ref{welastic}).
\begin{table}[h]
\caption{\label{tab:eigenmoment1}Extrapolated energy spectra $W_{n}$
using the model (\protect\ref{fitln}), resulting momenta $k_n$, and S-wave scattering phase
shifts on lattices $12^{3}\times24$ (upper table) and $10^{3}\times24$ (lower table).
Missing entries for $a_{t}k_n$ and $\delta_0(k_n)$ correspond to energy
levels outside of the elastic region (\protect\ref{welastic}).}
\begin{ruledtabular}
\begin{tabular}{clll}
$n$ & $a_{t}W_{n} $ & $a_{t}k_n$ & $\delta_0(k_n)$ \\
\colrule
1 & 0.64(16) & 0.16(1) & 74.6(11.6) \\
2 & 0.63(25) & 0.15(2) & 65.8(14.1) \\
3 & 0.19(5)  & -- & -- \\
4 & 0.47(11) & 0.03(1) & 2.1(1.8)   \\
5 & 0.31(11) & -- & -- \\
6 & 0.26(5)  & -- & -- \\
\colrule
1 & 0.66(21) & 0.20(2) & 77.1(10.5) \\
2 & 0.61(3)  & 0.16(1) & 50.0(6.1)  \\
3 & 0.49(4)  & 0.06(1) & 6.0(3.1)   \\
4 & 0.44(10) & 0.01(1) & 0.2(0.5)   \\
5 & 0.36(10)  & -- & -- \\
6 & 0.33(4)  & -- & -- \\
\end{tabular}
\end{ruledtabular}
\end{table}

The phase shift data are very sparse and do not alone resolve the functional
form of the fit model, such as Breit-Wigner. In fact, attempts of Levenberg-Marquardt
fits using (\ref{BWmodel})
only returned stable results for the resonance energy parameter $a_t E_0$, while the
width parameter $a_t \Gamma$, being an indicator for a derivative,
was left undetermined due to large standard errors.
Nevertheless, it is evident from Tab.~\ref{tab:eigenmoment1} that the phase shift
data are clustered around two regions of $a_t k_n$,
namely $\approx 0.15$--$0.16$ and $\approx 0.03$ for the $L=12$ lattice,
and $\approx 0.16$--$0.20$ and $\approx 0.01$--$0.06$ for the $L=10$ lattice.
This suggests that no more than two distinct physical states are uncovered by the
simulation. Under this assumption the data may be analyzed as follows:
Denoting the weighted ($\chi^2={\rm min}$) averages for each of the clustered momenta
by $\bar{k}_{1,2}$, the corresponding energies by $\omega_{1,2}= a_t W(\bar{k}_{1,2})$,
and $\tau_{1,2}={\rm tan}\,\delta_0(\bar{k}_{1,2})$, we obtain a set of two equations
from (\ref{BWmodel}), for each lattice, which are solved exactly by
\begin{eqnarray}
\label{gamma}
a_t \Gamma &=& 2(\omega_1-\omega_2)\frac {\tau_1 \tau_2}{\tau_1+\tau_2} \\
\label{E0}
a_t E_0 &=&\frac{\omega_2 \tau_2-\omega_1 \tau_1}{\tau_2-\tau_1}\,.
\end{eqnarray}
The resulting parameter values are listed in Tab.~\ref{tab:results}.
\begin{table}[b]
\caption{\label{tab:results}Results for decay widths $\Gamma$ and resonant energies
$E_0$ for two lattice sizes $L$ in units of $a_t$, and with a physical scale set by
the $\rho$ meson mass.}
\begin{ruledtabular}
\begin{tabular}{ccccc}
$L$ & $a_t\Gamma$ & $a_tE_0$ & $\Gamma[{\rm MeV}]$ & $E_0[{\rm GeV}]$\\
\colrule
12 &0.012(9)& 0.63(3)&35(26)&1.88(8)\\
10 &0.032(14)&0.62(3)&97(43)&1.84(9)\\
\end{tabular}
\end{ruledtabular}
\end{table}
There, the uncertainties for $a_t \Gamma$ and $a_t E_0$ are computed as follows:
The statistical (jackknife) errors for $a_t k_n$, as they appear in
Tab.~\ref{tab:eigenmoment1}, give rise to errors
$\triangle\bar{k}_{1,2}$ for the weighted momentum averages $\bar{k}_{1,2}$.
Repeating the analysis procedure described above a few thousand times with momenta
$k_{1,2}=\bar{k}_{1,2}+\xi\Delta \bar{k}_{1,2}$,
where $\xi$ is a normal distributed random deviate with variance one, then yields the
uncertainties given in Tab.~\ref{tab:results}.
Also the $a_1$ meson mass, which enters the fit model via (\ref{BWmodel1}), was
subjected to the same randomization. The errors given in Table \ref{tab:results} are the
standard deviations resulting from the randomization, they are reminiscent of
statistical errors. Table \ref{tab:results} also
contains the physical values for the decay widths and the resonance energies using the
$\rho$ meson to set the mass scale.
Setting the scale with the $a_1$ meson mass, results in widths of 39(29)MeV
and 108(48)MeV for the $12^3 \times 24$ and $10^3 \times 24$ lattices respectively.

Finally, Fig.~\ref{scaleindep} shows the combined phase shift
data from the two lattices on a common physical scale.
The curves are Breit-Wigner interpolations as explained above.
\begin{figure}[h]
\includegraphics[width=82mm,angle=0]{delta0_kGeV.eps}
\caption{\label{scaleindep}Scattering phase shifts $\delta_0(k_{n})$ from
the lattices $12^3 \times 24$ (filled circles) and $10^3 \times 24$ (open circles).
The solid and dashed curves are Breit-Wigner interpolations according to
(\protect\ref{gamma}) and (\protect\ref{E0}). The filled and open box plot symbols
indicate the respective $\chi^2$-weighted averages over data points.}
\end{figure}

\section{Errors}

All errors cited in this paper are statistical, and are derived from a standard jackknife
procedure \cite{Efr79}. The hybrid meson operator, involving gauge link paths,
appears to be the major source of those.

The effects of systematic errors on the results of this simulation
are difficult to assess. In principle this can only be done by repeating it with various different
choices of lattice and analysis model parameters. Probably the largest sources
of systematic error stem from curve fitting
and extrapolation techniques. As a check on the extrapolations we have also done fits with
a $x^{3/2}$ term in place of the logarithmic term in (\ref{fitln}).
The results, Tab.~\ref{tab:results}, did not change much, within statistical errors.
If only linear terms are retained the extrapolated masses slightly shift upward, ultimately
resulting in a slight increase of decay widths on the order of $\approx 10$MeV.

Another source of systematic error comes from postulating a Breit-Wigner model.
Given the sparsity
of data points it is inconclusive that the physical phase shifts will indeed follow a
Breit-Wigner form. In order to resolve this problem the simulation
would have to be repeated at several
values of the gauge coupling $\beta$, thus mapping out some sort of continuous curve $\delta_0(k)$
vs. $k$. The results of this work do rely on the {\it a priori} assumption
that the simulation data follow a Breit-Wigner model.

On the other hand, adopting the less stringent criterion that a resonance is present
if the phase shift data passes through $90^{\circ}$, the simulation results clearly
indicate the presence of such. This, in itself, is a significant outcome of this
project. Although this does not help putting bounds on the systematic error of
$\Gamma$, the results for the resonance energy, $E_0\approx 1.9$ GeV,
are remarkably stable. A decay width, on the other hand, essentially comes from
derivative data and as such is prone to a significantly larger error.

Systematic errors are also caused by finite size effects. At first sight, judging by
the small difference of the $\rho$ meson masses on the $L=12$ and $L=10$ lattices, see
Table \ref{tab:hopping}, those appear to be small. Finite size effects should be expected
to be much larger for larger-sized hadrons like the $a_1$ for example. This is particularly true
for two-hadron systems studied in this work. For example, the spectra displayed
in Fig.~\ref{spectra}
are significantly different, particularly for excited states on the $L=12$ and $L=10$ lattices,
their size though being quite similar. Again these effects can only be studied by repeating
this simulation with several lattices of different sizes.

Obtaining a single scattering phase data point
requires evaluating up to three effective mass functions - one for the correlator matrix eigenvalue,
one for the $\rho$ meson to set the physical scale, and one for the $\pi$ meson.
The variability in choosing which time slices of the correlation functions
to use in fitting effective masses produces a variability in the decay width.
Here, one usually wants to maximize the plateau width of the effective mass functions
to optimize the statistical error. Reducing the plateau width to estimate a systematic
error is of limited value.

\section{Conclusion}

Decay widths for the hybrid exotic meson with $J^{PC}=1^{-+}$, calculated using L\"{u}scher's method,
are in the range 35 to 97 MeV with statistical
errors of about 30 MeV using the $\rho$ meson to set the scale. The lower value for the width came
from using extrapolated energy spectra on a
$12^3\times 24$ lattice
and the higher value came from using extrapolated spectra on a $10^3\times 24$ lattice. If the $a_1$
meson sets the scale, then the widths for these two lattices range from 39 to 108 MeV with
statistical errors of about 40 MeV.

The number of data points available to fit Breit-Wigner functions is very sparse, the reason being
that many energy levels fell outside the elastic region where phase shifts using L\"{u}scher's formulae
cannot be computed. Overcoming this limitation requires use of a larger correlation matrix. This can
be accomplished by adding more smearing levels, and spatially extended operators in the individual
correlators, or possibly by using a coupled channel type analysis in which more than one decay
channel is represented in the matrix. Several values of the coupling parameter $\beta$ should also
be employed to generate more phase shift data points.

Using the $\rho$ meson to set the scale, the resonance mass of the hybrid meson in this
simulation was 1.9(1)~GeV, and in contrast to the decay width, the resonance mass was
well determined
by the simulation. This unexpected result leads to a final comment that, historically, hadron
mass calculations within lattice QCD have been done using single-hadron operators, ignoring
the fact that most hadrons are resonances and thus are unstable \cite{Michael:2005}. We
have taken this decay aspect seriously. Although the numerical values for
the decay widths serve as a guide only, the approach of extracting hadron masses as resonance energies,
 using L\"{u}scher's method, should also be given serious consideration.

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction}
Local operations and classical communication (LOCC) are an important subclass of operations on multipartite quantum systems, introduced first in the context of quantum information theory \cite{nielsen2010,diosi2011}. In the bipartite case, two subsystems are possessed by Alice and Bob respectively. They can carry out local operations as well as classically communicate with each other but are not allowed to exchange quantum information (\eg by sending each other photons in definite entangled states). The interest of this class of operations is that it can correlate the local subsystems but it cannot entangle them. For this reason, LOCC are instrumental in basic quantum information protocols, allowing entanglement to be seen as a resource one can gather (entanglement distillation) to subsequently enable tasks (such as quantum teleportation). Despite what their simple definition would suggest, quantum dynamical maps obtained from LOCC have a rich and intricate structure \cite{chitambar2014}. The time continuous setting, which we shall discuss below, has been less studied but already points to many interesting problems.

The motivation for the analysis of continuous time LOCC dynamics is unexpected and arose recently in a context a priori far removed from quantum information theory. Recent works \cite{kafri2014,kafri2015,tilloy2016,altamirano2016,tilloy2017,tilloy2017grw,bose2017} have indeed used continuous time LOCC to construct and constrain semi-classical theories of Newtonian gravity. The idea is that if the gravitational force is classical at a fundamental level, then perhaps it cannot be used to entangle particles. A sufficient condition\footnote{This is only a sufficient condition because there could exist non-entangling dynamics that cannot be simply derived from continuous time LOCC (see \eg the slightly larger class of semi-classical dynamics discussed in \cite{tilloy2017}). Further, the very definition of classical or non-quantum is sufficiently flexible that one may wish to extend it beyond non-entangling dynamics \cite{hall2017}.} is then that gravity can be implemented with LOCC. This provides a way to construct explicit theories of semi-classical gravity that do not require a modification of the statistical toolbox of quantum mechanics contrary to non-linear mean field approaches.

Let us informally define continuous time LOCC dynamics in a simple bipartite setting. In the discrete setting, Alice performs a local measurement on her system and broadcasts the measurement outcome to Bob. Knowing this outcome, Bob then performs a local unitary. These steps can be repeated arbitrarily many times while Alice and Bob can exchange their role. There is a natural continuous time limit to such an iteration of LOCC and one can obtain it using the theory of continuous quantum measurement. Alice continuously measures (or \emph{monitors}) a local observable $\hat{A}$ and broadcasts the time dependent measurement signal $a(t)$ to Bob. Bob then applies in real time a local feedback Hamiltonian $\Vo = a(t)\times \Bo$, proportional to the received signal. This is an instance of Markovian measurement based feedback in which the measured operators are restricted to a single subsystem of a bipartite decomposition. The corresponding Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) master equation \cite{gorini1976,lindblad1976}, even without this locality constraint, was derived a long time ago \cite{caves1987,diosi1990,wiseman1993,wiseman1994,diosi1994,wiseman2009}. Importantly, the feedback master equation contains non-local Hamiltonian terms in addition to local decoherence and possibly non-local dissipation.

This brings a number of natural questions. \emph{What kind of non-local potential can one construct this way? What is the price to pay in local decoherence or equivalently in local classical noise? Is there a natural way to define the minimum amount of noise required to implement a given non-local quantum potential with LOCC dynamics?} Our objective is to answer those questions in the slightly restrictive context of bipartite systems and for translation invariant potentials.

\section{Continuous local monitoring and non local feedback}

\subsection{One-way case}
We consider a bipartite quantum system of state $\hat{\rho}$ with each part under the control of Alice and Bob respectively. In the absence of feedback, we assume that the total Hamiltonian is a sum of local terms $\Ho=\Ho_A+\Ho_B$. We further assume that Alice is monitoring a set of observables $\{\Ao_\nu\}$. The statistics of the corresponding measurement signals $a_\nu(t)$ as well as the resulting backaction on the state can be written down using continuous measurement theory (see \eg \cite{jacobs2006,wiseman2009}). One gets:
\begin{equation}
a_\nu(t)=\Tr(\Ao_\nu\ro_t)+w^A_\nu(t),
\end{equation}
where the $w^A_\nu(t)$ are correlated white noises of zero average characterized by their two point function
\begin{equation}
\Ev\left[w^A_\nu(t) w^A_\mu(s)\right]=(\gamma^A)^{-1}_{\nu\mu}\delta(t-s).
\end{equation}
The real non-negative matrix $\gamma^A$ encodes the precision of Alice's monitoring device. Under this continuous measurement, the state $\ro^w_t$ evolves according to the stochastic master equation (SME):
\begin{equation}
    \frac{\upd\ro^w}{\upd t}=-i[\Ho_A+\Ho_B,\ro^w]
-\frac{1}{2}\sum_{\nu\mu}\left(\frac{\gamma^A_{\nu\mu}}{4}\left[\Ao_\nu,[\Ao_\mu,\ro^w]\right] - \gamma^A_{\nu\mu} \left\{\Ao_\nu - \langle \Ao_\nu\rangle_t, \ro^w_t\right\} w^A_\mu(t) \right)
\end{equation}
where $\langle \Ao_\nu\rangle_t = \Tr [\Ao_\nu\ro^w_t]$ and the multiplicative white noise is understood in the It\^o convention. If nothing is done with the measurement signals and if one averages over them, one gets the simple GKLS master equation for $\ro=\mathds{E}[\ro^w]$:
\begin{equation}
    \frac{\upd\ro}{\upd t}=-i[\Ho_A+\Ho_B,\ro]
-\frac{1}{2}\sum_{\nu\mu}\frac{\gamma^A_{\nu\mu}}{4}\left[\Ao_\nu,[\Ao_\mu,\ro]\right].
\end{equation}
However, if Bob introduces a local potential $\Vo_B(t)$ proportional to the instantaneous signals he receives, \ie
\begin{equation}
\Vo_B(t)=\sum_\nu a_\nu(t)\Bo_\nu,
\end{equation}
where $\{\Bo_\nu\}$ are local observables of Bob,
one can show that the noise-averaged state of the composite system satisfies the
GKLS equation:
\begin{equation}\label{eq:masterasym}
\frac{\upd\ro}{\upd t}=-i[\Ho_A+\Ho_B,\ro]
-\frac{1}{2}\sum_{\nu\mu}\left(i\left[\Bo_\nu,\{\Ao_\nu,\ro\}\right]
+\frac{\gamma^A_{\nu\mu}}{4}\left[\Ao_\nu,[\Ao_\mu,\ro]\right]
+(\gamma^A)^{-1}_{\nu\mu}\left[\Bo_\nu,[\Bo_\mu,\ro]\right]\right).
\end{equation}
We shall not discuss the derivation of this master equation which contains minor subtleties and which can be found in the literature (see \eg \cite{wiseman1994,wiseman2009,tilloy2016}). Rather, we take \eqref{eq:masterasym} as the starting point of our exploration of continuous time LOCC dynamics. We note already an interesting feature of the ME \eqref{eq:masterasym}: the non-Hamiltonian part is non-local due to the terms $-\frac{i}{2}[\Bo_\nu,\{\Ao_\nu,\ro\}]$ although the equation is non-entangling by construction, as it is obtained from LOCC dynamics.

\subsection{Symmetric case}
We extend the previous setting by adding the possibility for Bob to measure and Alice to apply a potential proportional to the corresponding instantaneous signal. The general case can be obtained in the same way as before. We focus on a restricted symmetric case which is particularly interesting. Let us assume that Bob is monitoring the same set of observables $\{\Bo_\nu\}$ that he is using for feedback and that Alice is using the same set of observables $\{\Ao_\nu\}$ for feedback that she is using for monitoring. The corresponding GKLS master equation --after averaging over the measurement outcomes-- reads:
\begin{equation}\label{eq:mastersym}
\frac{\upd\ro}{\upd t}=-i[\Ho_A+\Ho_B+\sum_\nu\Ao_\nu\Bo_\nu,\ro]
-\sum_{\nu\mu}\left(\frac{\gamma^A_{\nu\mu}}{8}+\frac{(\gamma^B)^{-1}_{\nu\mu}}{2}\right)\left[\Ao_\nu,[\Ao_\mu,\ro]\right]
-\sum_{\nu\mu}\left(\frac{\gamma^B_{\nu\mu}}{8}+\frac{(\gamma^A)^{-1}_{\nu\mu}}{2}\right)\left[\Bo_\nu,[\Bo_\mu,\ro]\right],
\end{equation}
where the only freedom lies in the two non-negative real matrices $\gamma^A$ and $\gamma^B$ setting the precision of Alice's and Bob's monitoring. The master equation \eqref{eq:mastersym} possesses two remarkable features. As before, it is obtained from LOCC and thus it does not entangle the two parties. Second, it contains a non-local Hamiltonian term:
\begin{equation}\label{eq:H_AB}
\Ho_{AB}=\sum_\nu\Ao_\nu\Bo_\nu.
\end{equation}
The coexistence of these two properties is made possible by the presence of local decoherence on both sides. Because every interaction Hamiltonian can be decomposed into a sum of tensor products of local terms, we answer the first question mentioned in the introduction: provided one accepts to pay a certain price in local decoherence, one can implement any non-local Hamiltonian on a bipartite system\footnote{The only subtlety, as we shall later see, is that implementing a given Hamiltonian may require divergent local decoherence terms making the corresponding master equation trivial.}. We now discuss more precisely the cost in local decoherence to implement a given pair potential between two particles.

\section{Implementing a pair potential}

\subsection{Setup}\label{sec:setup}
We assume that Alice and Bob each have a particle and that the two particles interact via a translation invariant pair potential $V(\xo_A-\xo_B)$ depending only on the particle coordinates. Naturally, even for initially disentangled states, the pair-potential will entangle the bipartite system. If, however, there is sufficiently strong local decoherence, then entanglement vanishes in finite time and the two particles remain only classically correlated \cite{diosi2003,yu2004}. We assume that the bipartite state $\hat{\rho}$ obeys the GKLS equation:
\begin{equation}\label{eq:localdeco}
\frac{\upd\ro}{\upd t}=-i\left[\Ho_A+\Ho_B+V(\xo_A-\xo_B),\ro\right]+\Dcal_A\ro+\Dcal_B\ro,
\end{equation}
where $\Ho_A$, $\Ho_B$ are local Hamiltonians and $\mathcal{D}_A$, $\mathcal{D}_B$ are local GKLS ``dissipators''. In light of the previous section, the natural question we may want to answer is the following. What structure and ``intensity'' need the local dissipator have to prevent the pair potential from entangling the two particles? A strategy to obtain a candidate is to construct a continuous time LOCC dynamics giving rise to a master equation of the form \eqref{eq:localdeco} and then to minimize the local dissipators.

We first write the pair potential as a sum of tensor products of local operators by Fourier expanding it. Because the potential is self-adjoint, we get:
\begin{align}
    V(\xo_A-\xo_B) &= \int_{\mathds{R}^3} \upd \kb\; v_\kb \cos\left[\kb\cdot(\xo_A-\xo_B)\right]\\
    &=\int_{\mathds{R}^3} \upd \kb\; v_\kb \left(\cos(\kb\cdot\xo_A)\cos(\kb\cdot\xo_B)+\sin(\kb\cdot\xo_A)\sin(\kb\cdot\xo_B)\right)
\end{align}
where $v_\kb$ is real. Defining now:
\begin{equation}\label{AksBks}
\Ao_{\kb,s}=\sqrt{|v_\kb|}\times\left\{
\begin{array}{c}
\cos(\kb\cdot\xo_A)~~~~s=1\\
\sin(\kb\cdot\xo_A)~~~~s=2
\end{array}
\right.
~~~~~~~~
\Bo_{\kb,s}=\mathrm{sgn}(v_\kb)\sqrt{|v_\kb|}\times\left\{
\begin{array}{c}
\cos(\kb\cdot\xo_B)~~~~s=1\\
\sin(\kb\cdot\xo_B)~~~~s=2
\end{array}
\right. ,
\end{equation}
we can identify the potential with the interaction Hamiltonian \eqref{eq:H_AB} of \eqref{eq:mastersym}:
\begin{equation}\label{V_expansion}
V(\xo_A-\xo_B)=\sum_{s=1,2}\int_{\mathds{R}^3} \upd \kb \; \Ao_{\kb,s}\Bo_{\kb,s}.
\end{equation}
Alice's dissipator reads:
\begin{equation}
\Dcal_A\ro=-\sum_{s,s'=1,2}\iint \upd \kb\upd \kb'\left(\frac{\gamma^A_{\kb,s,\kb',s'}}{8}+\frac{(\gamma^B)^{-1}_{\kb,s,\kb',s'}}{2}\right)\left[\Ao_{\kb,s},[\Ao_{\kb',s'},\ro]\right],
\end{equation}
and Bob's can be written in the same way by exchanging $A$ and $B$.

\subsection{Principle of least decoherence}
To find a way to canonically minimize local decoherence, we make the assumption that the dissipators in the final GKLS equation have the same symmetries as the potential, \ie that they are translation invariant and that they have the $A\leftrightarrow B$ symmetry. This drastically reduces the freedom in the precision matrix which can be written:
\begin{equation}
\gamma^A_{\kb,s,\kb',s'} = \gamma^B_{\kb,s,\kb',s'} = \gamma_{\kb}\,\delta_{s,s'}\,\delta^3(\kb-\kb').
\end{equation}
With this simplification, Alice's dissipator becomes:
\begin{equation}
\Dcal_A\ro=-\sum_{s=1,2}\int \upd \kb\left(\frac{\gamma_\kb}{8}+\frac{1}{2\gamma_\kb}\right)\left[\Ao_{\kb,s},[\Ao_{\kb,s},\ro]\right].
\end{equation}
We now have a clear principle of least decoherence (analogous the the one already discussed in \cite{tilloy2017}) which consists in minimizing all the coefficients independently for each wave number. This yields $\gamma_\kb=2$. In the end the Alice's local decoherence term reads:
\begin{align}
    \mathcal{D}_A \ro&=-\frac{1}{2} \int \upd \kb \,|v_\kb| \left(\left[\cos(\kb\cdot \xo_A),[\cos(\kb\cdot \xo_A),\hat{\rho}]\right]+\left[\sin(\kb\cdot \xo_A),[\sin(\kb\cdot \xo_A),\hat{\rho}]\right]\right)\\
    &= \iint \mathds{V}(\xb-\xb') \delta^3(\xb-\xo_A)\, \ro\,  \delta^3(\xb'-\xo_A) \,\upd \xb\upd \xb' -\mathds{V}(0)\,\ro, \label{eq:dissipator}
\end{align}
where $\mathds{V}$ is the potential with $|v_\kb|$ as Fourier modes (instead of $v_\kb$ for the initial pair potential). If $V$ is a non-negative kernel (as is the case for most non-relativistic Green's functions), then $\mathds{V}=V$. This is a remarkable result: the minimum of decoherence is reached when the spatial local decoherence functional is given by the pair-potential. This brings a severe difficulty for pair-potentials diverging in $0$ (as is the case for the Newtonian potential) as the resulting minimum decoherence is then infinite. Implementing such a potential in an LOCC way requires a regularization at short distances to keep decoherence finite (see \cite{tilloy2017} for a discussion in the context of gravity).

\subsection{Minimal disentangling local noise}
The previous results apply in a slightly different context than the one of \ref{sec:setup}. Consider now that he two particles are subjected to local noisy potentials in addition to the entangling pair potential. We assume that their evolution is given by the following stochastic Schr\"odinger equation:
\begin{equation}\label{eq:sse}
\frac{\upd}{\upd t} \ket{\psi}=
-i\left(\Ho_A+\Ho_B+V(\xo_A-\xo_B)+\xi_A(\xo_A,t)+\xi_B(\xo_B,t)\right)\ket{\psi}
\end{equation}
where $\xi_A$ and $\xi_B$ are independent white Gaussian noises of zero mean with the same two point function
\begin{equation}
\Ev\left[\xi_A(x,t)\xi_A(x',t')\right]=\Ev\left[\xi_B(x,t)\xi_B(x',t')\right]=D(x-x')\delta(t-t'),
\end{equation}
and where the multiplicative noise is understood in the Stratonovich convention\footnote{This convention is natural in this context because the noise is physical. Physically, equation \eqref{eq:sse} is obtained in the limit where the time correlation of the noise is much smaller than all other timescales. By virtue of the Wong-Zakaï theorem \cite{wong1965}, this limit corresponds to the stochastic differential equation in Stratonovich form.}. The only constraint on $D$ is that it is a real function corresponding to a non-negative kernel (that is, $D$ has positive Fourier transform). We may ask what amount of local noise is sufficient to destroy the entangling properties of the pair potential. This can be done simply by noting that the GKLS master equation obtained from \eqref{eq:sse}, after averaging, has local dissipators of the same form as those of equation \eqref{eq:dissipator}. Indeed, writing $\ro=\mathds{E}[\ket{\psi}\bra{\psi}]$ one can show\footnote{This can be done by rewriting \eqref{eq:sse} in the It\^o convention, using It\^o's lemma to get a stochastic master equation for $\ket{\psi}\bra{\psi}$, and finally removing the It\^o integral by averaging. Equivalently one may solve \eqref{eq:sse} by formal exponentiation in the Stratonovich representation, average over the noise by Gaussian integration, and differentiate the result.} that $\ro$ obeys the GKLS master equation \eqref{eq:localdeco} with:
\begin{equation}
     \mathcal{D}_A \ro= \iint D(\xb-\xb') \delta^3(\xb-\xo_A)\, \ro\,  \delta^3(\xb'-\xo_A) \,\upd \xb\upd \xb' -D(0)\,\ro,
\end{equation}
and the same for $B$. Identifying this expression with \eqref{eq:dissipator} we naturally fix $D=\mathds{V}$, \ie the correlation function of the local noises is given by the pair potential (or a version with positive Fourier transform were it not non-negative).

This gives us the noise threshold for which the stochastic Schr\"odinger equation \eqref{eq:sse} is non-entangling (on average). As an aside, this provides a simple demonstration that many-body dynamics are easy to simulate classically provided a sufficient amount of local classical noise is added.

\section{Summary}

We have shown that it is possible to create any potential --not necessarily harmonic-- between two particles using LOCC. The corresponding GKLS master equations possess local decoherence terms that take a particularly simple form provided a principle of least decoherence is applied. In that case the local decoherence functional is simply equal to the pair potential (if the latter has a non-negative Fourier transform).

\begin{acknowledgments}
L.D. thanks the Hungarian Scientific Research Fund under
Grant No. 124351, and the EU COST Actions MP1209 and CA15220 for support. A.T. thanks the Alexander von Humboldt foundation and the Agence Nationale de la Recherche (ANR) contract ANR-14-CE25-0003-01 for support.
\end{acknowledgments}