text:
\section{Introduction}

Let $\hilb$ be a Hilbert space. In what follows, ${\mathbb B}(\hilb)$
denotes the $\sp{\ast}$-algebra of all bounded linear operators on
$\hilb$, and ${\mathbb G}(\hilb)\subseteq{\mathbb B}(\hilb)$ is
the subgroup of the additive group of ${\mathbb B}(\hilb)$ consisting
of all Hermitian operators on $\hilb$.  The additive group
${\mathbb G}(\hilb)$, organized into a partially ordered abelian group
as usual, will be called the \emph{Hermitian group} for $\hilb$. The
identity operator ${\mathbf 1}$ belongs to ${\mathbb G}(\hilb)$ and
satisfies ${\mathbf 0}\leq{\mathbf 1}$.  We define ${\mathbb E}
(\hilb) :=\{E\in{\mathbb G}(\hilb)\mid{\mathbf 0}\leq E\leq{\mathbf 1}\}$
and (following G. Ludwig \cite{Lud}) we refer to operators in
${\mathbb E}(\hilb)$ as \emph{effect operators} on $\hilb$. We also
define ${\mathbb P}(\hilb) :=\{P\in{\mathbb G}(\hilb)\mid P\sp{2}=P\}$
to be the set of all (orthogonal) \emph{projection operators} on $\hilb$.
Then we have
\[
{\mathbf 0},{\mathbf 1}\in{\mathbb P}(\hilb)\subseteq{\mathbb E}(\hilb)
 \subseteq{\mathbb G}(\hilb)\subseteq{\mathbb B}(\hilb).
\]

In mathematical physics, the representation of observables by
so-called POV-measures, i.e., ${\mathbb E}(\hilb)$-valued measures on
$\sigma$-fields \cite{BLM}, as well as by the more conventional
${\mathbb P}(\hilb)$-valued measures, has now become a commonplace.
Consequently, ${\mathbb E}(\hilb)$, ${\mathbb P}(\hilb)$, and
suitable generalizations thereof, have come to be employed as
algebraic models for the semantics of both sharp and unsharp
quantum logics \cite{CMW,DGG,DP,FGB,GF,GPBB,LM}. Thus motivated,
we take the pair $({\mathbb B}(\hilb),{\mathbb E}(\hilb))$
consisting of the ring ${\mathbb B}(\hilb)$ and the ``effect algebra"
${\mathbb E}(\hilb)$ as a prototype for the following more general
notion of an ``e-ring."

\begin{definition} \label{def:eoring}
\textnormal{An \emph{e-ring} is a pair $(R,E)$ consisting of an
associative ring $R$ with unity $1$ and a subset $E\subseteq R$ of
elements called \emph{effects} such that $0,1\in E$; $e\in E
\Rightarrow 1-e\in E$; and the set $E\sp{+}$ consisting of all finite
sums $e\sb{1}+e\sb{2}+\cdots+e\sb{n}$ with $e\sb{1},e\sb{2},...,e\sb{n}
\in E$ satisfies the following conditions: For all $a,b\in E\sp{+}$;}\\

\begin{tabular}{ll}
{\rm (i)} $-a\in E\sp{+}\Rightarrow a=0$, &
{\rm (ii)} \ $1-a\in E\sp{+}\Rightarrow a\in E$,\\
{\rm (iii)} $ab=ba\Rightarrow ab\in E\sp{+}$, &
{\rm (iv)} $aba\in E\sp{+}$,\\
{\rm (v)} \ $aba=0\Rightarrow ab=ba=0$, and &
{\rm (vi)}  $(a-b)\sp{2}\in E\sp{+}$.
\end{tabular}\\

\end{definition}

The notion of an e-ring is mathematically equivalent to the notion
of an effect-ordered ring originally introduced in \cite{FCPOAG}, but
reformulated to emphasize the role of the ``effect algebra" $E$.

It is easy to see that the prototypic pair $({\mathbb B}(\hilb),
{\mathbb E}(\hilb))$ is an e-ring as soon as one observes that
${\mathbb E}(\hilb)\sp{+}$ is precisely the set of all positive
Hermitian operators on $\hilb$. (By a slight abuse of language, we
call a Hermitian operator $A$ ``positive" if $A\geq{\mathbf 0}$.)
Indeed, every effect operator is positive by definition, and a finite
sum of positive Hermitian operators is positive. Conversely, if $A$
is a positive Hermitian operator, $\|A\|$ is the uniform operator
norm of $A$, and $n$ is a positive integer with $n\geq\|A\|$, then
$(1/n)A$ is an effect operator, and $A$ is a sum of $n$ effect
operators, each equal to $(1/n)A$.

In addition to the prototypic example $({\mathbb B}(\hilb),
{\mathbb E}(\hilb))$, we have the following simple examples
of e-rings. (More examples will be given later.) We denote the
set of positive integers by ${\mathbb N} :=\{1,2,3,...\}$.

\begin{example} \label{ex:Eis01}
Let $R$ be a ring with unity $1$ such that $n\cdot 1\not=0$ for all
$n\in{\mathbb N}$, and define $E :=\{0,1\}$. Then $(R,E)$ is an e-ring.
\end{example}

\begin{example} \label{ex:subfield}
If $R$ is any subfield of the totally ordered field $\reals$ of real
numbers and $E :=\{e\in R\mid 0\leq e\leq 1\}$, then $(R,E)$ is an
e-ring.
\end{example}

As Examples \ref{ex:Eis01} and \ref{ex:subfield} illustrate,
the definition of an e-ring $(R,E)$ does not necessarily provide
a strong connection between the algebraic structure of the set
$E$ of effects and the ring structure of $R$.  For instance, if
$(R,E)$ is an e-ring and ${\widetilde R}$ is any extension ring
of $R$ such that $1\cdot x=x\cdot 1=x$ for all $x\in{\widetilde R}$,
then $({\widetilde R},E)$ is also an e-ring. The rather weak
connection between $E$ and $R$ does not concern us here because,
motivated by quantum measurement theory and quantum logic, we are
mainly interested in the structure of the ``effect algebra" $E$,
and the enveloping ring $R$ is just a convenient environment
in which to study this structure.

Section 2 below, which treats the basic properties of e-rings,
culminates with a theorem that the set $P$ of projections in an e-ring
acquires (at least) the structure of an orthomodular poset
(Theorem \ref{th:OMP}) and a theorem stating that $P$ indexes a
so-called compression base for the directed group generated by the
effects (Theorem \ref{th:compbase}). Section 3, which treats
commutativity and coexistence in the effect algebra $E$ of an e-ring,
culminates in a structure theorem for a class of e-rings (called
b-rings) in which every effect is a projection (Theorem
\ref{th:bring}). In a subsequent paper, we shall study square
roots and polar decompositions in e-rings.

\section{Basic Properties of e-Rings}

We begin by extracting from an e-ring $(R,E)$ an analogue $G$
of the Hermitian group ${\mathbb G}(\hilb)$ for the prototype
$({\mathbb B}(\hilb),{\mathbb E}(\hilb))$.

\begin{theorem} \label{th:G}
Let $(R,E)$ be an e-ring and let $E\sp{+}$ be the set of all
sums of finite sequences f elements of $E$. Then
\[
G :=E\sp{+}-E\sp{+}=\{a-b\mid a,b\in E\sp{+}\}
\]
is a subgroup of the additive group of the ring $R$, and $G$
is a directed partially ordered abelian group with positive
cone $E\sp{+}=\{g\in G\mid 0\leq g\}$. Moreover, $E\subseteq G$
and $E$ generates the group $G$.
\end{theorem}

\begin{proof}
Since $E\sp{+}$ is closed under addition, it is clear that
$G=E\sp{+}-E\sp{+}$ is a subgroup of the additive group of
the ring $R$.  Also, $0\in E\subseteq E\sp{+}\subseteq G$,
and by Definition \ref{def:eoring} (i), if both $a$ and $-a$
belong to $E\sp{+}$, then $a=0$. Therefore, $G$ is a partially
ordered abelian group with positive cone $E\sp{+}$, the partial
order being given by $g\leq h\Leftrightarrow h-g\in E\sp{+}$ for
$g,h\in G$ \cite[p. 3]{Good}. By the definition of $G$, every
element $g\in G$ can be written in the form $g=a-b$ with $a,b
\in E\sp{+}$, i.e., $G$ is directed \cite[p. 4]{Good}. Thus,
$E\sp{+}$ generates the group $G$, and since $E$ generates
$E\sp{+}$ as an additive semigroup, it follows that $E$
generates $G$ as a group.
\end{proof}

If $(R,E)$ is an e-ring then, \emph{as an additive abelian group},
$R$ is partially ordered (but not necessarily directed) with
$E\sp{+}$ as its positive cone; however, unless $R$ is commutative,
it is not necessarily a partially ordered ring (as usually
understood) because $E\sp{+}$ need not be closed under multiplication.

\begin{definition} \label{def:GandP}
\textnormal{Let $(R,E)$ be an e-ring and let $E\sp{+}$ be the set
of all sums of finite sequences of elements of $E$.  Then:}
\begin{enumerate}

\item\textnormal{The partially ordered additive abelian group $G :=
 E\sp{+}-E\sp{+}$ (Theorem \ref{th:G}) is called the \emph{directed
group} of $(R,E)$.}
\item\textnormal{Idempotent elements $p=p\sp{2}\in G$ are called
 \emph{projections}.}

\end{enumerate}
\end{definition}

For the prototype e-ring $({\mathbb B}(\hilb),{\mathbb E}(\hilb))$,
the directed group is the Hermitian group ${\mathbb G}(\hilb)$, and
${\mathbb P}(\hilb)$ is the set of projections. In Example
\ref{ex:Eis01}, the directed group $G=\{n\cdot 1\mid n\in\integers\}$
of $(R,E)$ is isomorphic to the totally ordered additive group of the
ring $\integers$ of integers. In Example \ref{ex:subfield}, $G$ is
the additive subgroup of the field $R$ with the total order inherited
from $\reals$. In both Examples \ref{ex:Eis01} and \ref{ex:subfield},
the only projections are $0$ and $1$.

The e-ring $({\mathbb B}(\hilb),{\mathbb E}(\hilb))$ is a special
case (when $A$ is a type-I von Neumann factor) of the e-ring $(A,E)$
in the following example.

\begin{example} \label{ex:CStar}
Let $A$ be a unital C\,$\sp{\ast}$-algebra and let
\[
E :=\{aa\sp{\ast}\mid a\in A\text{ and }\exists\,b\in A, aa\sp{\ast}
 +bb\sp{\ast}=1\}.
\]
Then $(A,E)$ is an e-ring, $E\sp{+}=\{aa\sp{\ast}\mid a\in A\}$,
and the directed group $G$ for $(A,E)$ is the additive group of
self-adjoint elements in $A$.
\end{example}

\noindent{\bf In the sequel, we assume that $(R,E)$ is an e-ring,
$1$ is the unity element in $R$, $E\sp{+}$ is the set of all sums
of finite sequences of elements of $E$, $G=E\sp{+}-E\sp{+}$ is
the directed group of $(R,E)$, $G$ is partially ordered with
positive cone $E\sp{+}$, and
\[
P=\{p\in G\mid p=p\sp{2}\}
\]
is the set of all projections in $G$. It is understood that $P$ and
$E$ are partially ordered by the restrictions of the partial order
$\leq$ on $G$.}

\begin{lemma} \label{lm:AA} Let $g,h\in G$ and let $p\in P$.
Then:
\smallskip

\begin{tabular}{lll}
{\rm (i)} \ \ $g\sp{2}\in E\sp{+}$.  &
{\rm (ii)} \  $gh+hg\in G$. &
{\rm (iii)} $g\in E\sp{+}\Rightarrow ghg\in G$. \\
{\rm (iv)} $php\in G$. &
{\rm (v)} \  $h\in E\sp{+}\Rightarrow php\in E\sp{+}$.
\end{tabular}

\end{lemma}

\begin{proof} (i) By hypothesis, $G$ is directed, hence $g$
is a difference of two elements in $E\sp{+}$, so $g\sp{2}\in
E\sp{+}$ by Definition \ref{def:eoring} (vi).

(ii) By (i), $gh+hg=(g+h)\sp{2}-g\sp{2}-h\sp{2}\in G$.

(iii) As $G$ is directed, there exist $a,b\in E\sp{+}$ such
that $h=a-b$. By Definition \ref{def:eoring} (iv), $gag,gbg
\in E\sp{+}$, whence $ghg=gag-gbg\in G$.

(iv) As $p=p\sp{2}\in G$, (i) implies that $p\in E\sp{+}$;
 hence (iv) follows from (iii).

(v) As in (iv), $p\in E\sp{+}$, so (v) follows from Definition
\ref{def:eoring} (iv).
\end{proof}

\begin{lemma} \label{lm:E}
{\rm (i)} $E=\{e\in G\mid 0\leq e\leq 1\}$. {\rm (ii)} $0,1\in P$. {\rm (iii)}
$p\in P\Rightarrow 1-p\in P$. {\rm (iii)} $P\subseteq E$.
\end{lemma}

\begin{proof}
(i) By Definition \ref{def:eoring}, we have $0,1\in E$. Evidently,
if $e\in E$, then $e,1-e\in E\subseteq E\sp{+}=\{g\in G\mid 0\leq g\}$,
whence $e\in G$ with $0\leq e\leq 1$. Conversely, suppose $e\in G$
with $0\leq e\leq 1$. Then $e,1-e\in E\sp{+}$, and it follows from
Definition \ref{def:eoring} (ii) that $e\in E$. Thus, $E=\{e\in G
\mid 0\leq e\leq 1\}$.

(ii) $0=0\sp{2}\in G$ and $1=1\sp{2}\in G$.

(iii) If $p\in G$ with $p=p\sp{2}$, then $1-p\in G$ with
$(1-p)\sp{2}=1-2p+p\sp{2}=1-p$.

(iv) Suppose $p\in P$. By Lemma \ref{lm:AA} (i), $p=p\sp{2}\in
E\sp{+}$, i.e., $0\leq p$. Thus, by (iii), $0\leq 1-p$, whence
$0\leq p\leq 1$, so $p\in E$ by (i).
\end{proof}

In view of Lemma \ref{lm:E} (i), we shall refer to $E$ as the
\emph{unit interval} in $G$. We have
\[
0,1\in P\subseteq E\subseteq E\sp{+}\subseteq G\subseteq R.
\]
Equipped with the partially defined binary operation $\oplus$
obtained by restricting $+$ on $G$ to $E$, the unit interval $E$
forms a so-called \emph{effect algebra} \cite{BF97}.  The
effect algebras arising from e-rings in this way are rather special
in that they admit a (perhaps only partial) multiplicative structure
(cf. \cite{D02}).

\begin{lemma} \label{lm:FF}
Let $d,e,f\in E$ with $ef=fe$. Then:\smallskip\\
\begin{tabular}{ll}
{\rm (i)} \ \ $0\leq ef\leq e,f\leq 1$. &
{\rm (ii)} $0\leq ede\leq e\sp{2}\leq e\leq 1$. \smallskip\\
{\rm (iii)} $0\leq e,f\leq e+f-ef\leq 1$. &
{\rm (iv)} $0\leq e-e\sp{2}\leq e,1-e\leq 1$.
\end{tabular}
\end{lemma}

\begin{proof} Assume the hypotheses.

(i) By Definition \ref{def:eoring} (iii), $0\leq ef$. Likewise,
$0\leq e(1-f)=e-ef$, so $ef\leq e$, and by symmetry,
$ef\leq f$.

(ii) By Definition \ref{def:eoring} (iv), $0\leq ede$ and
$0\leq e(1-d)e=e\sp{2}-ede$.  Also, by (i) with $f=e$, we have
$e\sp{2}\leq e\leq 1$.

(iii) By (i), $0\leq f-ef$, so $e\leq e+f-ef$, and by symmetry,
$f\leq e+f-ef$. Also, by Definition \ref{def:eoring} (iii),
$0\leq(1-e)(1-f)=1-e-f+ef$, whence $e+f-ef\leq 1$.

(iv) By (ii), $0\leq e-e\sp{2}$. Also, by Lemma \ref{lm:AA}
(i), $0\leq e\sp{2}$, whence $e-e\sp{2}\leq e$. Finally, by (iii)
with $f=e$, we have $2e-e\sp{2}\leq 1$, so $e-e\sp{2}\leq 1-e$.
\end{proof}

\begin{lemma} \label{lm:M}
Let $g,h,k\in E\sp{+}$, $p\in P$, and $n\in{\mathbb N}$. Then:
{\rm (i)} $gh=0\Rightarrow hg=0$. {\rm (ii)} If
$gk=kg$ and $hk=kh$, then $g\leq h\Rightarrow gk\leq hk$.
{\rm (iii)} If $gh=hg$, then $g\leq h\Rightarrow g\sp{2}
\leq h\sp{2}$. {\rm (iv)} $g\leq np\Rightarrow g=gp=pg$.
{\rm (v)} $g\sp{n}=0\Rightarrow g=0$.
\end{lemma}

\begin{proof}
(i) $gh=0\Rightarrow ghg=0\Rightarrow hg=0$ by Definition
\ref{def:eoring} (v).

(ii) Assume the hypotheses. Then $h-g\in E\sp{+}$ and
$hk-gk=(h-g)k=k(h-g)\in E\sp{+}$ by Definition \ref{def:eoring}
(iii).

(iii) Assume the hypotheses. By (ii), $g\sp{2}\leq gh$ and
$gh\leq h\sp{2}$, so $g\sp{2}\leq h\sp{2}$.

(iv) Assume the hypotheses. Then $g,np-g\in E\sp{+}$, whence
$(1-p)g(1-p),(1-p)(np-g)(1-p)=-(1-p)g(1-p)\in E\sp{+}$ by
Lemma \ref{lm:AA} (v).  Therefore, $(1-p)g(1-p)=0$ by
Definition \ref{def:eoring} (i), and it follows from Definition
\ref{def:eoring} (v) that $(1-p)g=g(1-p)=0$, i.e., $g=pg=gp$.

(v) We may assume that $n$ is the smallest positive integer such
that $g\sp{n}=0$. If $n$ is even and $k=n/2$, we have $g\sp{k}
\cdot 1\cdot g\sp{k}=0$, so $g\sp{k}=g\sp{k}\cdot 1=0$ by
Definition \ref{def:eoring} (v), contradicting our assumption.
Therefore $n$ is odd. If $n=1$, we are done, so we may assume that
$n=2k+1$ where $k\in{\mathbb N}$. Then $g\sp{k}gg\sp{k}=0$, so $g\sp{k+1}
=g\sp{k}g=0$ by Definition \ref{def:eoring} (v), again contradicting
our assumption.
\end{proof}

According to part (i) of the following lemma, $1$ is a so-called
\emph{order unit} in $G$ \cite[p. 4]{Good}.

\begin{lemma} \label{lm:orderunit}
{\rm (i)} If $g\in G$, there exists $n\in{\mathbb N}$ such that $g\leq n
\cdot 1$. {\rm (ii)} If $a\sb{1},a\sb{2},...,a\sb{n}\in E\sp{+}$
and $a\sb{1}+ a\sb{2}+\cdots+a\sb{n}=0$, then $a\sb{1}=a\sb{2}=\cdots
=a\sb{n}=0$.
\end{lemma}

\begin{proof}
(i) Write $g=a-b$ with $0\leq a,b$. Then $0\leq b=a-g$, whence
$g\leq a$. As $a\in E\sp{+}$, there exist $e\sb{1},e\sb{2},
...,e\sb{n}\in E$ with $a=e\sb{1}+e\sb{2}+\cdots+e\sb{n}$.
By Lemma \ref{lm:E} (i), $e\sb{i}\leq 1$ for $i=1,2,...,n$,
and it follows that $g\leq a\leq n\cdot 1$.

(ii) Assume the hypotheses. It will be sufficient to prove that
$a\sb{1}=0$. But, $-a\sb{1}=a\sb{2}+\cdots+a\sb{n}\in E\sp{+}$,
so $a\sb{1}=0$ by Definition \ref{def:eoring} (i).
\end{proof}

\begin{theorem} \label{th:CC}
Let $e\in E$ and $p\in P$. Then the following conditions are
mutually equivalent: {\rm (i)} $e\leq p$, {\rm (ii)} $e=ep=pe$,
{\rm (iii)} $e=pep$, {\rm (iv)} $e=ep$, {\rm (v)} $e=pe$.
\end{theorem}

\begin{proof}
(i) $\Rightarrow$ (ii). Assume that $e\leq p$ and let $d :=p-e$.
Then $e,d\in E\sp{+}$, $e+d=p$, and
\[
(1-p)e(1-p)+(1-p)d(1-p)=(1-p)p(1-p)=0.
\]
By Lemma \ref{lm:E} (iii), $1-p\in P$, whence by Lemma \ref{lm:AA}
(v),
\[
(1-p)e(1-p),(1-p)d(1-p)\in E\sp{+},
\]
and it follows from Lemma \ref{lm:orderunit} (ii) that $(1-p)e(1-p)=(1-p)d(1-p)
=0$. Therefore, by Definition \ref{def:eoring} (v), $(1-p)e=e(1-p)=0$,
i.e., $e=pe=ep$.

(ii) $\Rightarrow$ (iii) $\Rightarrow$ (iv). Follows from $p=p\sp{2}$.

(iv) $\Leftrightarrow$ (v). By Lemma \ref{lm:M} (i),
$e=pe\Rightarrow (1-p)e=0\Rightarrow e(1-p)=0\Rightarrow e=ep$,
and the converse implication follows by symmetry.

(v) $\Rightarrow$ (i).  Assume (v). Since (iv)
$\Leftrightarrow$ (v), we have $pe=ep=e$, so $(1-e)p=p(1-e)
=p-e$, whence  $p-e\in E\sp{+}$ by Definition \ref{def:eoring}
(iii), and therefore $e\leq p$.
\end{proof}

\begin{corollary} \label{cor:DD}
Let $e\in E$ and $p\in P$. Then the following conditions are
mutually equivalent: {\rm (i)} $p\leq e$, {\rm (ii)} $p=ep=pe$,
{\rm (iii)} $p+pep=pe+ep$, {\rm (iv)} $p=ep$, {\rm (v)} $p=pe$.
\end{corollary}

\begin{proof}
Replace $e$ by $1-e$ and $p$ by $1-p$ in Theorem \ref{th:CC}, noting
that $p\leq e\Leftrightarrow 1-e\leq 1-p$.
\end{proof}

\begin{theorem} \label{th:p+qinP}
Let $p,q\in P$.  Then the following conditions are mutually
equivalent: {\rm (i)} $p+q\in E$, {\rm (ii)} $p+q\leq 1$,
{\rm (iii)} $pq=0$, {\rm (iv)} $pq=qp=0$, {\rm (v)}
$p+q\in P$.
\end{theorem}

\begin{proof}
(i) $\Rightarrow$ (ii) Follows from Lemma \ref{lm:E} (i).

(ii) $\Rightarrow$ (iii). If $p+q\leq 1$, then $p\leq 1-q$,
and it follows from Theorem \ref{th:CC} that $p=p(1-q)=p-pq$,
whence $pq=0$.

(iii) $\Rightarrow$ (iv). Follows from Lemma \ref{lm:M} (i).

(iv) $\Rightarrow$ (v). If $pq=qp=0$, then $(p+q)\sp{2}=
p\sp{2}+pq+qp+q\sp{2}=p+q$.

(v) $\Rightarrow$ (i). Follows from Lemma \ref{lm:E} (iv).
\end{proof}

\begin{theorem} \label{th:pqinP}
If $p,q\in P$, then the following conditions are
mutually equivalent: {\rm (i)} $pq\in P$, {\rm (ii)} $pq\in
E$, {\rm (iii)} $pq=qp$, {\rm (iv)} $pq=pqp$.
Moreover, if any---hence all---of these conditions hold,
then $pq=p\wedge q$ is the infimum {\rm (}greatest lower
bound{\rm )} of $p$ and $q$ both in $P$ and in $E$.
\end{theorem}

\begin{proof}
(i) $\Rightarrow$ (ii) is obvious.

(ii) $\Rightarrow$ (iii). Assume that $e :=pq\in E$. Since
$e=pe=eq$, Theorem \ref{th:CC} implies that $e\leq p,q$
and $e=ep=qe$.  Thus, $e\sp{2}=epq=e$, so $e\in P$. As
$e\leq p,q$, it follows that $p-e,q-e\in E\sp{+}$.
Furthermore, $(p-e)(q-e)=pq-pe-eq+e\sp{2}=0$,
whence Lemma \ref{lm:M} (i) implies that
$0=(q-e)(p-e)=qp-e$, i.e., $qp=e=pq$.

(iii) $\Rightarrow$ (iv). If $pq=qp$, then $pq=p(pq)=pqp$.

(iv) $\Rightarrow$ (i). If $pq=pqp$, then $(pq)\sp{2}=
pqpq=(pq)q=pq$, so $pq\in P$.

Suppose that any, hence all of the conditions (i)--(iv) hold.
Then $pq\in P$ with $pq\leq p,q$. Also, if $e\in E$ with
$e\leq p,q$, then $e=ep=eq$, whence $e(pq)=e$, i.e.,
$e\leq pq$.
\end{proof}

\begin{corollary} \label{cor:pqinP}
If $p,q\in P$ and $pq=qp$, then $p+q-pq\in P$ and
$p+q-pq=p\vee q$ is the supremum {\rm (}least upper bound{\rm )}
of $p$ and $q$ both in $P$ and in $E$.
\end{corollary}

\begin{proof}
The mapping $r\mapsto(1-r)$ is order inverting and of period
$2$ on $P$. Also $pq=qp\Leftrightarrow(1-p)(1-q)=(1-q)(1-p)$,
and $1-(1-p)(1-q)=p+q-pq$.
\end{proof}

\begin{corollary} \label{cor:q-pinP}
Let $p,q\in P$. Then $q-p\in E\Leftrightarrow p\leq q
\Leftrightarrow q-p\in P$. Moreover, if $p\leq q$, then
$q-p=q\wedge(1-p)$ is the infimum of $q$ and $1-p$ both
in $P$ and in $E$.
\end{corollary}

\begin{proof}
If $q-p\in E$, then $0\leq q-p$, whence $p\leq q$.
Suppose that $p\leq q$. Then $p=pq=qp$ by Theorem \ref{th:CC},
so $q(1-p)=(1-p)q=q-p$ and it follows from Theorem \ref{th:pqinP}
that $q-p\in P$ and that $q-p$ is the infimum both in $E$
and $P$ of $q$ and $1-p$.
\end{proof}

\begin{theorem} \label{th:OMP}
With $p\mapsto 1-p$ as orthocomplementation, $P$ is an
orthomodular poset {\rm (OMP) \cite{DGG,DP}} and, for $p,q\in P$
with $p\leq 1-q$, the supremum in $P$ of $p$ and $q$
is $p\vee q=p+q$.
\end{theorem}

\begin{proof}
We have $0\leq p\leq 1$ for all $p\in P$, and $p\mapsto 1-p$ is
an order-reversing mapping of period $2$ on $P$. Let $p,q\in P$.
If $p\leq 1-q$, then $p+q\leq 1$, whence $p+q\in E$, so
$p+q\in P$ with $pq=qp=0$ by Theorem \ref{th:p+qinP}, and it
follows from Corollary \ref{cor:pqinP} that $p+q=p+q-pq$ is the
supremum of $p$ and $q$ in $P$. Now suppose that $p\leq q$.
By Corollary \ref{cor:q-pinP}, $q-p\in P$, whence $q=p+(q-p)$
is the orthomodular identity.
\end{proof}

\begin{lemma} \label{lm:Pnormal}
Suppose that $d,e,f,d+e+f\in E$ with $d+e,d+f\in P$. Then $d,e,f\in P$.
\end{lemma}

\begin{proof}
Assume the hypotheses and let $p :=d+e\in P$ and $q :=d+f\in P$.
Then $p+f=d+e+f\leq 1$, so $f\leq 1-p\in P$; hence by Theorem
\ref{th:CC}, $f=(1-p)f=f-pf$, and therefore $pf=0$. Also, $d\leq
d+e=p$, so $pd=d$ by Theorem \ref{th:CC}. Consequently, $pq=
p(d+f)=d\in E$, and it follows from Theorem \ref{th:pqinP} that
$d=pq=pq\in P$. As $d\leq p$ and $d,p\in P$, it follows from
Corollary \ref{cor:q-pinP} that $e=p-d\in P$, and likewise $f=
q-d\in P$.
\end{proof}

The following theorem provides useful conditions---not directly
involving multiplication---for determining whether an effect
is a projection.

\begin{theorem} \label{th:sharp}
If $e\in E$, then the following conditions are mutually equivalent:
{\rm (i)} If $a,b,a+b\in E$, then $a,b\leq e\Rightarrow a+b\leq e$.
{\rm (ii)} If $d\in E$ with $d\leq e,1-e$, then $d=0$. {\rm (iii)}
$e\in P$.
\end{theorem}

\begin{proof}
(i) $\Rightarrow$ (ii). Assume (i) and the hypotheses of (ii).
Then $d+e\leq 1$ with $d,e\leq e$, whence $d+e\leq e$, and it
follows that $d=0$.

(ii) $\Rightarrow$ (iii). Assume (ii).  By Lemma \ref{lm:FF} (iv),
$0\leq e-e\sp{2}\leq e,1-e,$ so $e-e\sp{2}=0$, i.e., $e=e\sp{2}$.

(iii) $\Rightarrow$ (i).  Suppose $e\in P$ and assume the
hypotheses of (i). By Corollary \ref{cor:DD}, $a=ae$ and $b=be$,
whence $(a+b)e=a+b$, and it follows that $a+b\leq e$.
\end{proof}

If $p\in P$ and $g\in G$, then by Lemma \ref{lm:AA} (iv), $pgp\in G$;
hence we can define the mapping $J\sb{p}\colon G\to G$ by
$J\sb{p}(g)=pgp$ for all $g\in G$.  Thus, owing to Lemmas \ref{lm:AA},
\ref{lm:E}, \ref{lm:Pnormal}, and Theorem \ref{th:p+qinP}, we have
the following theorem (see \cite{FDenver,FCB}).

\begin{theorem} \label{th:compbase}
The family $(J\sb{p})\sb{p\in P}$ is a compression base for $G$.
\end{theorem}

The partially ordered abelian group $G$ is said to be
\emph{archimedean} iff, whenever $g,h\in G$ and $ng\leq h$
for all $n\in{\mathbb N}$, it follows that $g\leq 0$ \cite[p. 20]{Good}.
An order-preserving group endomorphism $J\colon G\to G$ is
called a \emph{retraction} iff $J(1)\in E$ and, for all $e\in E$,
$e\leq J(1)\Rightarrow J(e)=e$. If $p\in P$, it is clear that
$J\sb{p}$ is a retraction on $G$. Conversely, as a consequence
of \cite[Corollary 4.6]{FCPOAG}, we have the following theorem.

\begin{theorem}
If $G$ is archimedean, then every retraction $J$ on $G$ has the
form $J=J\sb{p}$ with $p=J(1)\in P$.
\end{theorem}

\section{Commuting Elements of $G$}

\emph{We maintain our standing hypothesis that $(R,E)$ is an e-ring,
$G$ is its directed group, and $P$ is the OMP of projections
in $G$.}

\begin{definition}
\textnormal{Let $g,h\in G$.  We write $gCh$ iff $gh=hg$ and we define
the \emph{commutant} of $g$ in $G$ by $C(g) :=\{h\in G\mid gCh\}$.
More generally, if $X\subseteq G$, then $C(X) :=\bigcap\sb{x\in X}C(x)$
is called the \emph{commutant} of $X$.}
\end{definition}

In contrast with more-or-less standard usage, e.g., in operator theory,
we use the notion of the commutant \emph{only in relation to
elements of} $G$, and not to general elements of the enveloping
ring $R$.

If $L$ is any OMP, then two elements $p,q\in L$ are said to be
\emph{Mackey compatible} iff there exist pairwise orthogonal
elements $p\sb{1},q\sb{1},d\in L$ with $p=d\vee p\sb{1}$, and
$q=d\vee q\sb{1}$ \cite{DP}. By Theorem \ref{th:OMP}, projections $p,q$
in the OMP $P$ are Mackey compatible iff there exist projections
$p\sb{1},q\sb{1},d\in P$ with $d+p\sb{1}+q\sb{1}\leq 1$,
$p=d+p\sb{1}$, and $q=d+q\sb{1}$. The next lemma provides a useful
condition---not directly involving multiplication---for
determining whether two projections commute.

\begin{lemma} \label{lm:pCq}
If $p,q\in P$, then $pCq$ iff $p$ and $q$ are Mackey compatible in
$P$.
\end{lemma}

\begin{proof}
Suppose that $pCq$. By Theorem \ref{th:pqinP}, $pq\in P$ with
$pq\leq p,q$; by Corollary \ref{cor:pqinP}, $p+q-pq\in P$;
and by Corollary \ref{cor:q-pinP}, $p\sb{1} :=p-pq\in P$ and
$q\sb{1} :=q-pq\in P$.  Thus, with $d :=pq$, we have $d+
p\sb{1}+q\sb{1}=p+q-pq\in P$ with $p=d+p\sb{1}$, and $q=d+q\sb{1}$.

Conversely, suppose there exist $p\sb{1},q\sb{1},d\in P$
such that $d+p\sb{1}+q\sb{1}\in P$, $p=d+p\sb{1}$, and
$q=d+q\sb{1}$. Then $d+p\sb{1}\leq d+p\sb{1}+q\sb{1}\leq 1$,
whence $dCp\sb{1}$ by Theorem \ref{th:p+qinP}. Likewise,
$dCq\sb{1}$, and $p\sb{1}Cq\sb{1}$, whence $pCq$.
\end{proof}

In the following definition, the condition in Lemma \ref{lm:pCq}
is generalized to effects $e,f\in E$.

\begin{definition}
\textnormal{Effects $e,f\in E$ are said to be \emph{coexistent} iff
there exist effects $d,e\sb{1},f\sb{1}\in E$ such that $d+e\sb{1}+f\sb{1}
\in E$, $e=d+e\sb{1}$, and $f=d+f\sb{1}$.}
\end{definition}

The terminology ``coexistent" is borrowed from the quantum theory of
measurement \cite{BLM}. (Some authors also refer to coexistent effects
as being ``Mackey compatible," but, since coexistent effects need not
commute, we prefer not to follow this practice.)

\begin{lemma} \label{lm:CimpliesCE}
Let $e,f\in E$. Then: {\rm (i)} If $eCf$, then $e$ and $f$ are
coexistent. {\rm (ii)} If $e+f\leq 1$, then $e$ and $f$ are coexistent.
\end{lemma}

\begin{proof}
(i) Let $d :=ef$, $e\sb{1} :=e-ef$, and $f\sb{1} :=f-ef$. By Lemma
\ref{lm:FF} (i), $d,e\sb{1},f\sb{1}\in E$. Also, $d+e\sb{1}+f\sb{1}=
e+f-ef\in E$ by Lemma \ref{lm:FF} (iii).

(ii) If $e + f\in E$, then $0+e+f\in E$ with $e=0+e$ and $f=0+f$.
\end{proof}

In general, the converse of Lemma \ref{lm:CimpliesCE} (i) is false.
For instance, in the prototype ${\mathbb E}(\hilb)$, choose two
effect operators $A$ and $B$ that do not commute. Then $\frac{1}{2}A$
and $\frac{1}{2}B$ are non-commuting effect operators; yet, since
$\frac{1}{2}A+\frac{1}{2}B\leq{\mathbf 1}$, they are coexistent.
However, we do have the following result.

\begin{theorem}
Let $p,q\in P$. Then, regarded as effects in $E$, the projections
$p$ and $q$ are coexistent iff $pCq$.
\end{theorem}

\begin{proof}
Combine Lemma \ref{lm:Pnormal} and Lemma \ref{lm:pCq}.
\end{proof}

In a Boolean algebra (i.e., a bounded complemented distributive
lattice), every element has a unique complement; hence
if an OMP is a Boolean algebra, the Boolean complementation
mapping coincides with the orthocomplementation mapping.
It is well known that an OMP is a Boolean algebra iff its
elements are pairwise Mackey compatible \cite{DP}; hence
we have the following.

\begin{corollary} \label{cor:PBoo}
The OMP $P$ is a Boolean algebra iff $P\subseteq C(P)$.
Moreover, if $P$ is a Boolean algebra, then $p\mapsto
1-p$ is the Boolean complementation mapping on $P$.
\end{corollary}

In what follows, we shall be considering the condition
$G\subseteq C(G)$ and the weaker condition $G\subseteq C(P)$.
For instance, if the enveloping ring $R$ is commutative, then
$G\subseteq C(G)$. Since $E$ generates the group $G$, it follows
that $G\subseteq C(G)\Leftrightarrow E\subseteq C(E)$ and that
$G\subseteq C(P)\Leftrightarrow E\subseteq C(P)$. Also, if
$G\subseteq C(P)$, then $P\subseteq C(P)$, whence $P$ is a
Boolean algebra by Corollary \ref{cor:PBoo}. If the unital
C$\sp{\ast}$-algebra $A$ in Example \ref{ex:CStar} satisfies
virtually any version of the spectral theorem (e.g., if $A$ is a
von Neumann algebra, or even an AW$\sp{\ast}$-algebra), then
$P\subseteq C(P)$ will imply that $G\subseteq C(G)$.

Suppose that $G\subseteq C(G)$, let $g,h\in G$, and choose
$a,b,c,d\in E\sp{+}$ such that $g=a-b$ and $h=c-d$. By Definition
\ref{def:eoring} (iii), $ac,ad,bc,bd\in E\sp{+}\subseteq G$,
whence $gh=ac-ad-bc+bd\in G$, and it follows that $G$ is not only an
additive abelian group, but a commutative subring of $R$. Clearly,
with $G$ thus organized into a ring, $(G,E)$ is an e-ring, $G$ is a
partially ordered commutative ring with unity $1$, the partially
ordered additive group $G$ is the directed group of $(G,E)$,
and $(G,E)$ is a \emph{c-ring} as per the following definition.

\begin{definition}  \label{def:cring}
\textnormal{A \emph{c-ring} is an e-ring $(G,E)$ such that $G$ is
a commutative ring and $G=E\sp{+}-E\sp{+}$.}
\end{definition}

If $G\subseteq C(G)$ and $R\not=G$, \emph{we can disregard the
enveloping ring $R$ and drop down to the c-ring $(G,E)$.} Evidently,
the passage from the e-ring $(R,E)$ to the c-ring $(G,E)$ affects
neither the structure of the effect algebra $E$ nor of the Boolean
algebra $P$.

As a consequence of the Gelfand representation theorem \cite
[Theorem 4.4.3]{KR}, the following example of a c-ring may be
regarded as the commutative version of Example \ref{ex:CStar}.

\begin{example} \label{ex:CofX}
Let $X$ be a compact Hausdorff space, define $C(X,\reals)$ to be
the ring of all continuous real-valued functions $f\colon X\to\reals$
with pointwise operations, and let
\[
E(X,\reals) :=\{e\in C(X,\reals)\mid 0\leq e(x)\leq 1, \forall x
 \in X\}.
\] Then $(C(X,\reals),E(X,\reals))$ is a c-ring, the partial order
on $C(X,\reals)$ is the pointwise partial order, $C(X,\reals)$ is
archimedean, and the Boolean algebra
\[
P(X,\reals) :=\{p\in C(X,\reals)\mid p(x)\in
\{0,1\},\forall x\in X\}
\]
of projections in $(C(X,\reals),E(X,\reals))$ consists of the
characteristic set functions $\chi\sb{K}$ of compact open subsets
$K$ of $X$.
\end{example}

In the following example of a c-ring, the effects are ``fuzzy
subsets" of $X$ in the sense of L. Zadeh \cite{Zad}.

\begin{example} \label{ex:sigmafield}
Let ${\cal F}$ be a $\sigma$-field of subsets of a nonempty
set $X$, define ${\cal B}(X,{\cal F},\reals)$ to be the
ring under pointwise operations of all bounded real-valued
${\cal F}$-measurable functions $f\colon X\to\reals$, and let
\[
{\cal E}(X,{\cal F},\reals) :=\{e\in{\cal B}(X,{\cal F},
 \reals)\mid 0\leq e(x)\leq 1,\forall x\in X\}.
\] Then $({\cal B}(X,{\cal F},\reals),{\cal E}(X,{\cal F},
\reals))$ is a c-ring, the partial order on ${\cal B}
(X,{\cal F},\reals)$ is the pointwise partial order,
${\cal B}(X,{\cal F},\reals)$ is archimedean, and the
Boolean algebra
\[
{\cal P}(X,{\cal F},\reals) :=\{p\in{\cal B}(X,{\cal F},
 \reals)\mid p(x)\in\{0,1\},\forall x\in X\}
\]
of projections in ${\cal E}(X,{\cal F},\reals)$ consists of
the characteristic set functions $\chi\sb{M}$ of sets
$M\in{\cal F}$.
\end{example}

Recall that a partially ordered abelian group $G$ is said to be
\emph{lattice ordered}, or for short, is an \emph{$\ell$-group},
iff every pair of elements $g,h\in G$ has an infimum $g\wedge
\sb{G} h$ and a supremum $g\vee\sb{G} h$ in the partially
ordered set $G$. The additive partially ordered abelian
groups $C(X,\reals)$ and ${\cal B}(X,{\cal F},\reals)$ in
Examples \ref{ex:CofX} and \ref{ex:sigmafield} are $\ell$-groups
with pointwise minimum and maximum as the infimum and supremum,
respectively.

If $G$ has the property that, for every $a,b,c,d\in G$ with
$a,b\leq c,d$ (i.e., $a\leq c,a\leq d,b\leq c,$ and $b\leq d$),
there exists $t\in G$ with $a,b\leq t\leq c,d$, then $G$ has the
\emph{Riesz interpolation property}, or for short, $G$ is an
\emph{interpolation group} \cite[Chapter 2]{Good}. If $G$
is an $\ell$-group, then it is an interpolation group. (Just
take $t$ to be any element between $a\vee\sb{G}b$ and
$c\wedge\sb{G}d$.) Thus, the directed groups $C(X,\reals)$ and
${\cal B}(X,{\cal F})$ are interpolation groups.

The so-called \emph{MV-algebras}, which play an important role in
the analysis of many-valued logics \cite{CCC57,CCC58} and in the
classification of AF C$\sp{\ast}$-algebras \cite{Mund}, can be
characterized as the effect algebras that are realized as unit
intervals in abelian $\ell$-groups with order units. Thus, the unit
intervals $E(X,\reals)$ and ${\cal E}(X,{\cal F},\reals)$ in
Examples \ref{ex:CofX} and \ref{ex:sigmafield} are MV-algebras.
Not every MV-algebra can be realized as the unit interval in a
c-ring, but the author does not know a perspicuous
characterization of those that can.

In the theory of operator algebras, there are well-known
connections between commutativity and lattice structure. For
instance, by a theorem of S. Sherman \cite{Sherm}, a unital
C$\sp{\ast}$-algebra $A$ (Example \ref{ex:CStar}) is commutative
iff the directed group $G$ of self-adjoint elements in $A$ is an
$\ell$-group. On the other hand, by a result of R. Kadison \cite{Kad},
if $A$ is a von Neumann algebra, then $A$ is a factor iff the
directed group $G$ is an antilattice (i.e., only pairs of comparable
elements can have an infimum or a supremum in $G$). Under suitable
hypotheses (borrowed from the theory of operator algebras) similar
results can be obtained for groups with compression bases \cite{FoPu};
hence for e-rings. If $P$ is a Boolean algebra, then it is a lattice,
so Corollary \ref{cor:PBoo} already furnishes a hint of the
commutativity-lattice connection for e-rings; further
evidence is provided by Theorems \ref{th:ellBoo}, \ref{th:ellgroup},
and \ref{th:E=P} below.

\begin{theorem} \label{th:ellBoo}
Suppose that $G$ is an $\ell$-group {\rm (}or more generally, an
interpolation group {\rm \cite{Good})}. Then, {\rm (i)} $G\subseteq
C(P)$ and {\rm (ii)} $P$ is a Boolean algebra.
\end{theorem}

\begin{proof} Assume that $G$ is an interpolation group.

(i) Let $0\leq g\in G$ and $p\in P$. As $G$ is directed, it will be
sufficient to prove that $gCp$. By Lemma \ref{lm:orderunit} (i), there
exists $n\in{\mathbb N}$ such that $0\leq g\leq n\cdot 1$, whence
$0\leq g\leq np+n(1-p)$. As $G$ is an interpolation group, there
exist $x,y\in G$ with $0\leq x\leq np$, $0\leq y\leq n(1-p)$, and
$g=x+y$ (see \cite[Proposition 2.1 (b)]{Good}.  Thus, by Lemma
\ref{lm:M} (iv), $x=xp=px$ and $y=y(1-p)=(1-p)y$, and it follows
that $yp=py=0$ and $gp=pg=x$.

(ii) Follows from (i) and Corollary \ref{cor:PBoo}.
\end{proof}

The c-ring in Example \ref{ex:sigmafield} satisfies the conditions
in the following theorem. Of course, the c-ring in Example
\ref{ex:CofX} satisfies condition (i), and it satisfies condition
(ii) if $X$ is basically disconnected (i.e., the closure of every
open F$\sb{\sigma}$ subset of $X$ remains open).

\begin{theorem} \label{th:ellgroup}
Suppose that {\rm (i)} $G\subseteq C(P)$, and {\rm (ii)} for every
$g\in G$, there exists $p\in P$ with $(1-p)g\leq 0\leq pg$. Then $G$
is an $\ell$-group.
\end{theorem}

\begin{proof}
The proof is adapted from \cite[Proposition 8.9]{Good}. Let $g,h\in G$.
It will be sufficient to prove that the supremum  $g\vee\sb{G}h$ exists
in $G$. By hypothesis, there exists $p\in P$ such that $(1-p)(g-h)\leq 0
\leq p(g-h)$. Put
\[
s :=pg+(1-p)h=p(g-h)+h=(1-p)(h-g)+g.
\]
Evidently, $h,g\leq s$. Furthermore, if $k\in G$ with $g,h\leq k$,
then by Lemma \ref{lm:M} (i), $pg\leq pk$ and $(1-p)h\leq(1-p)k$,
whence $s\leq pk+(1-p)k=k$. Therefore, $s=g\vee\sb{G}h$.
\end{proof}

A subset $A\subseteq P$ is said to be \emph{orthogonal} iff,
for all $a,b\in A$, $a\not=b\Rightarrow ab=0$. If $A$ is an
orthogonal subset of $P$, then by Corollary \ref{cor:pqinP}
and induction on $n$, the sum $p :=a\sb{1}+a\sb{2}+\cdots+
a\sb{n}$ of finitely many distinct elements $a\sb{1},a\sb{2},
...,a\sb{n}\in A$ belongs to $P$ and coincides with the
supremum $p :=a\sb{1}\vee a\sb{2}\vee\cdots\vee a\sb{n}$
both in $P$ and in $E$.

\begin{theorem} \label{th:E=P}
The following conditions are mutually equivalent:
\begin{enumerate}
\item $P$ is a Boolean algebra and $P$ generates the group $G$.
\item $G\subseteq C(G)$ and, for each $g\in G$, there is a
 finite orthogonal set $A\subseteq P$ such that $g$ is a
 linear combination with integer coefficients of the elements of
 $A$.
\item $G$ is an $\ell$-group and $E=P$.
\item $G$ is an interpolation group and $1$ is a minimal order
 unit in $G$.
\item $E$ is a Boolean algebra with $e\mapsto 1-e$ as the
 Boolean complementation mapping.
\item $E=P$.
\end{enumerate}
\end{theorem}

\begin{proof}
(i) $\Rightarrow$ (ii). Assume (i). By Corollary \ref{cor:PBoo},
$P\subseteq C(P)$ and, since $P$ generates $G$, it follows that
$G\subseteq C(G)$.  Let $g\in G$. Then there are projections
$p\sb{i}\in P$, $1\leq i\leq n$, and integer coefficients
$c\sb{i}$ such that $g=\sum\sb{i=1}\sp{n}c\sb{i}p\sb{i}$.
Let $B$ be the sub-Boolean algebra of $P$ generated by $p\sb{i}$,
$1\leq i\leq n$. Since $B$ is a finitely generated Boolean
algebra, it is finite. Let $A$ be the set of atoms (minimal
nonzero elements) in $B$. Then, if $a,b\in A$ with $a\not=b$, we
have $ab=ba=a\wedge b=0$ (Theorem \ref{th:pqinP}), so $A$ is a
finite orthogonal subset of $P$. Also, each element in $B$, and
in particular each $p\sb{i}$, can be written as a sum of certain
of the projections in $A$.  Thus, by gathering terms, we can
write $g=\sum\sb{i=1}\sp{n}c\sb{i}p\sb{i}=\sum\sb{a\in A}k\sb{a}a$
with integer coefficients $k\sb{a}$ for all $a\in A$.

(ii) $\Rightarrow$ (iii). Assume (ii). Then $G\subseteq C(G)
\subseteq C(P)$. Let $g\in G$, and let $A$ be a finite orthogonal
subset of $P$ such that $g=\sum\sb{a\in A}k\sb{a}a$ for integer
coefficients $k\sb{a}$. Define $A\sb{+} :=\{a\in A\mid k\sb{a}>0\}$,
$A\sb{-} :=\{a\in A\mid k\sb{a}<0\}$, and $p :=\sum\sb{a\in A\sb{+}}a$.
Then $p\in P$, $a\in A\sb{+}\Rightarrow pa=k\sb{a}a$, and $a\in A\sb{-}
\Rightarrow pa=0$. Thus, $pg=\sum\sb{a\in A\sb{+}}k\sb{a}a\geq 0$ and
$(1-p)g=g-pg=\sum\sb{a\in A\sb{-}}k\sb{a}a\leq 0$; hence $G$ is an
$\ell$-group by Theorem \ref{th:ellgroup}. Now suppose $g\in E$.
Then, if $a\in A\sb{-}$, we have $0\leq ga=ag$, whence $0\leq ga=
k\sb{a}a\leq 0$, so $k\sb{a}a=0$. Consequently, $g=\sum\sb{a\in
A\sb{+}}k\sb{a}a$. Also, if $a\in A\sb{+}$, we have $a\leq k\sb{a}a
=ga\leq a$ (Lemma \ref{lm:FF} (i)), so $k\sb{a}a=a$. Consequently,
$g=\sum\sb{a\in A\sb{+}}k\sb{a}a=\sum\sb{a\in A\sb{+}}a=p\in P$.
Therefore, $E=P$.

(iii) $\Rightarrow$ (iv).  Assume (iii). Then $G$ is an interpolation
group. Suppose $p$ is an order unit in $G$ and $p\leq 1$. Then
$0\leq p$ and there exists $n\in{\mathbb N}$ such that $0\leq 1-p\leq np$.
As $0\leq p\leq 1$, we have $p\in E=P$; hence by Lemma \ref{lm:M} (iv),
$1-p=(1-p)p=0$, i.e., $p=1$.  Thus $1$ is a minimal order unit in $G$.

(iv) $\Rightarrow$ (v). Assume (iv). By Theorem \ref{th:ellBoo},
$G\subseteq C(P)$ and $P$ is a Boolean algebra with $p\mapsto 1-p$
as the Boolean complementation. It will be enough to show that
$E=P$. Thus, let $e\in E$ and suppose that $d\in E$ with $d\leq
e,1-e$. Then $1-e\leq 1-d$ and $e\leq 1-d$. Adding the last two
inequalities, we find that $1\leq 2(1-d)$. Thus, if $g\in G$, there
exists $n\in{\mathbb N}$ such that $g\leq n\cdot 1\leq 2n(1-d)$, and it
follows that $1-d$ is an order unit in $G$. But $1-d\leq 1$; hence,
by hypothesis, $1-d=1$, i.e., $d=0$. Consequently, $e\in P$ by
Theorem \ref{th:sharp}, and we conclude that $E=P$.

(v) $\Rightarrow$ (vi).  Assume (v) and let $e\in E$. If $d\in E$
with $d\leq e,1-e$, then since $1-e$ is the Boolean complement of
$e$ in the Boolean algebra $E$, it follows that $d=0$; hence
$e\in P$ by Theorem \ref{th:sharp}.  Consequently, $E=P$.

(vi) $\Rightarrow$ (i). Assume (vi) and let $p,q\in P=E$. Since $E$
generates the group $G$, it will be sufficient by Corollary \ref
{cor:PBoo} to prove that $pCq$.  Let $d :=pqp$.  By Lemma \ref{lm:FF}
(ii), $d\in E=P$, and it is clear that $d=dp=pd$, whence $d\leq p$.
Thus, $dqd=dpqpd=d\sp{3}=d$, and it follows that $d(1-q)d=0$.
Therefore, $d(1-q)=0$, i.e., $d=dq$, so $d\leq q$. By Corollary
\ref{cor:q-pinP}, $p\sb{1} :=p-d\in P$ and $q\sb{1} :=q-d\in P$.
Also, $pq\sb{1}p=pqp-pdp=d-d=0$, so $pq\sb{1}=0$, $p=d+p\sb{1}$,
$q=d+q\sb{1}$, and by Theorem \ref{th:p+qinP} $d+p\sb{1}+q\sb{1}
=p+q\sb{1}\in P$.  Consequently, $p$ and $q$ are Mackey compatible,
so $pCq$ by Lemma \ref{lm:pCq}.
\end{proof}

If the e-ring $(R,E)$ satisfies any, hence all, of the conditions
(i)--(v) in Theorem \ref{th:E=P}, then $G\subseteq C(G)$ by condition
(ii), and we can drop down to the c-ring $(G,E)$, which of course
will continue to satisfy conditions (i)--(v).

\begin{definition}
\textnormal{A \emph{b-ring} is a c-ring $(G,E)$ satisfying any,
hence all, of the conditions (i)--(v) in Theorem \ref{th:E=P}.}
\end{definition}

The b-ring in the following example is a modification of Example
\ref{ex:sigmafield} in which the totally ordered field $\reals$ is
replaced by the totally ordered ring $\integers$ of integers and
the $\sigma$-field ${\cal F}$ is replaced by any field of sets.

\begin{example} \label{ex:field}
Let ${\cal F}$ be a field of subsets of a nonempty set $X$, define
${\cal B}(X,{\cal F},\integers)$ to be the ring under pointwise
operations of all bounded functions $f\colon X\to\integers$ such that
$f\sp{-1}(z)\in{\cal F}$ for all $z\in\integers$, and let
\[
 {\cal E}(X,{\cal F},\integers) :=\{e\in{\cal B}(X,{\cal F},
 \integers)\mid e(x)\in\{0,1\},\forall x\in X\}.
\]
Then $({\cal B}(X,{\cal F},\integers),{\cal E}(X,{\cal F},\integers))$
is a b-ring, the partial order on ${\cal B}(X,{\cal F},\integers)$
is the pointwise partial order, and ${\cal B}(X,{\cal F},\integers)$
is archimedean. Thus the effects in ${\cal E}(X,{\cal F},\integers)$,
which coincide with the projections for the b-ring $({\cal B}(X,{\cal F},
\integers),\\{\cal E}(X,{\cal F},\integers))$, are the characteristic
set functions $\chi\sb{M}$ of sets $M\in{\cal F}$.
\end{example}

Under set-inclusion, a field ${\cal F}$ of subsets of a nonempty
set $X$ is a Boolean algebra, and in Example \ref{ex:field}, the
Boolean algebra ${\cal E}(X,{\cal F},\integers)$ is isomorphic to
${\cal F}$. By the Stone representation theorem, every Boolean
algebra $B$ is isomorphic to the field ${\cal F}$ of compact open
subsets of a compact Hausdorff totally-disconnected space $X$;
hence, \emph{every Boolean algebra can be realized as the Boolean
algebra of projections in a b-ring.}

The functions $f\colon X\to\integers$ in Example \ref{ex:field} can
be regarded as ``signed multisets" by thinking of $f(x)$ as the
``signed multiplicity of $x$ in $f$."  In \cite{TH}, T. Hailperin
suggests that, in contemporary algebraic terms, the true realization of
Boole's original ideas is not what is now called a Boolean algebra,
but rather it is an algebra of signed multisets forming a commutative
ring with unity and with no nonzero additive or multiplicative
nilpotents. Our b-ring $({\cal B}(X,{\cal F},\integers),{\cal E}
(X,{\cal F},\integers))$ is precisely such an algebra, and the
``b" in ``b-ring" is meant to suggest this Boolean connection.

\begin{theorem} \label{th:Booext}
Let $(G,E)$ and $(H,F)$ be b-rings and let $\phi\colon E\to F$ be a
Boolean homomorphism from the Boolean algebra $E$ to the Boolean
algebra $F$. Then $\phi$ admits a unique extension to a group
homomorphism $\Phi\colon G\to H$ of the additive group $G$ into
the additive group $H$.  Moreover, $\Phi\colon G\to H$ is an
order-preserving ring homomorphism with $\Phi(1)=1$.
\end{theorem}

\begin{proof}
The Boolean homomorphism $\phi\colon E\to F$ preserves $0$, $1$,
finite infima, and finite suprema. For $p,q\in E=P$, we have
$p\wedge q=pq$; hence $\phi(pq)=\phi(p\wedge q)=\phi(p)\wedge\phi(q)
=\phi(p)\phi(q)$, i.e., $\phi$ preserves products of projections.
Also, if $p+q\in E$, then $p\vee q=p+q$; hence $\phi(p+q)=\phi(p\vee q)
=\phi(p)\vee\phi(q)=\phi(p)+\phi(q)$, i.e., $\phi\colon E\to H$
preserves existing sums in $E$. Since $G$ is an interpolation group,
a theorem of S. Pulmannov\'{a} \cite{Pu99} implies that $\phi$ admits
a unique extension to a group homomorphism $\Phi\colon G\to H$. As
$\phi(E)\subseteq F$, it follows that $\Phi(E\sp{+})\subseteq F\sp{+}$,
whence $\Phi$ is order preserving.  Every element in $G$ is a finite
linear combination of projections with integer coefficients, and since
$\Phi$ preserves products of projections, it follows that $\Phi$
preserves products. Obviously, $\Phi(1)=\phi(1)=1$.
\end{proof}

The following is the fundamental structure theorem for b-rings.

\begin{theorem} \label{th:bring}
Let $(G,E)$ be a b-ring, let $X$ be the Stone space of the Boolean
algebra $E$, and let ${\cal F}$ be the field of compact open subsets
of $X$. Then there is an order and ring isomorphism $\Phi\colon G
\to{\cal B}(X,{\cal F},\integers)$ such that the restriction
$\phi$ of $\Phi$ to $E$ is a Boolean isomorphism of $E$ onto
${\cal E}(X,{\cal F},\integers)$.
\end{theorem}

\begin{proof}
The projections in ${\cal E}(X,{\cal F},\integers)$ are characteristic
set functions $\chi\sb{K}$ of compact open subsets $K$ of $X$; hence
by Stone's representation theorem, there is a Boolean isomorphism
$\phi\colon E\to{\cal E}(X,{\cal F},\integers)$. By Theorem
\ref{th:Booext}, $\phi$ can be extended to an order-preserving
ring homomorphism $\Phi\colon G\to{\cal B}(X,{\cal F},\integers)$
and $\phi\sp{-1}\colon{\cal E}(X,{\cal F},\integers)\to E$ can be
extended to an order-preserving ring homomorphism $\Psi\colon{\cal B}
(X,{\cal F},\integers)\to G$.  The ring endomorphism $\Psi\circ\Phi
\colon G\to G$ is the identity on $E$, and $E$ generates $G$; hence
$\Psi\circ\Phi\colon G\to G$ is the identity on $G$. Likewise
$\Phi\circ\Psi\colon$ is the identity on ${\cal B}(X,{\cal F},
\integers)$, so $\Phi$ is an order-preserving ring isomorphism
with $\Psi=\Phi\sp{-1}$.
\end{proof}


text:
\section{Overview and Motivation}
Almost for the past one decade, the AdS/CFT correspondence \cite{Maldacena:1997re}-\cite{Aharony:1999ti} has been found to provide an extremely elegant tool in order to explore various physical properties of strongly coupled (gauge theory) plasma at sufficiently high temperatures. The hydrodynamic description of such strongly coupled gauge theories has been studied quite successfully by considering asymptotically AdS black holes in the dual gravitational counterpart \cite{Policastro:2001yc}-\cite{Kovtun:2008kx}. The underlying motivation behind such analysis rests on the fact that the Quark Gluon Plasma (QGP) produced at RHIC, Brookhaven is strongly coupled where the usual techniques of perturbative Quantum Field Theory (QFT) do not apply.

Apart from being strongly coupled, the other characteristic feature of the QGP produced at RHIC is the anisotropic expansion of the fireball during the very early stage of the collision \cite{Ryblewski:2010bs}-\cite{Martinez:2010sd} which therefore has driven a lot of attention in the context of holography \cite{Mateos:2011ix}-\cite{Cheng:2014qia}. In \cite{Mateos:2011ix}-\cite{Mateos:2011tv}, the authors had proposed a systematic anisotropic construction in the context of Einstein-axion-dilaton gravity where they have considered a particular anisotropic ($ \theta $ deformed) version of $ \mathcal{N}=4 $ SYM plasma namely, $ \delta S_{YM}\sim \int \theta(z) Tr F\wedge F $, where the $ \theta $ parameter (which is dual to the axion in the bulk) depends linearly on one of the spatial directions of the brane. The corresponding hydrodynamic analysis of their model has been performed in \cite{Rebhan:2011vd}. The key outcomes of their analysis could be summarized as follows: (1) The DC conductivity along the isotropic direction of the brane is different from that of its value corresponding to the anisotropic direction, and most importantly, (2) the shear viscosity to entropy ($ \eta/s $) ratio corresponding to the longitudinal fluctuations has been found to differ significantly from that of its value computed from the transverse fluctuations. The most significant outcome of their analysis rests on the fact that one could have a natural violation of the conjectured lower bound on $ \eta/s $ ratio solely from the anisotropic considerations even in the context of Einstein gravity \cite{Rebhan:2011vd}.

Even before these analysis had performed, in \cite{Landsteiner:2007bd} the authors had studied hydrodynamics of a strongly coupled plasma in a slightly different context of anisotropy which was driven due to presence of the non commutativity along different spatial directions of the $ Dp $ brane in the presence of a background NS B field. Holographically such theories are supposed to describe non commutative $ \mathcal{N}=4 $ SYM plasma at strong coupling \cite{Seiberg:1999vs}-\cite{Hashimoto:1999ut}. In their analysis \cite{Landsteiner:2007bd}, the authors had found that despite of the spatial anisotropy (that is caused due to the distinction between the commutative and the non commutative spatial directions) the shear viscosity to entropy ($ \eta/s $) ratio turns out to be universal for two different shear channels. The reason that the universality of the bound is still maintained in the non commutative scenario could be understood in terms of the holographic stress tensor which surprisingly turns out to be the same as that of the commutative theory \cite{Landsteiner:2007bd}.

In summary, from the comparative analysis in the previous two paragraphs, one should be able to note that the $ \theta $ deformed $ \mathcal{N}=4 $ SYM differs significantly from that of the non commutative $ \mathcal{N}=4 $ SYM as long as we consider the hydrodynamic description of both the theories with respect to their shear channels. However, the comparison remains incomplete as the analysis of the diffusive modes, in particular the computation of the $ R $ charge diffusion corresponding to non commutative $ \mathcal{N}=4 $ SYM  theory is still lacking in the literature. The purpose of the present article is therefore to fill up this gap and make a systematic comparison between two different anisotropic theories at strong coupling. In order to do that we essentially turn on $ U(1) $ fluctuations in the bulk and compute the corresponding $ R $ charge diffusion rates along both the commutative as well as the non commutative directions of the brane. Unlike the case for the shear viscosity \cite{Landsteiner:2007bd}, we observe a significant deviation in the charge transport phenomena along the non commutative direction of the brane. On the other hand, the charge diffusion constant along the direction of the commutative coordinates of the brane does not receive any non commutative corrections and thereby remains unchanged.

The organization of the paper is the following: In Section 2, we discuss the geometrical construction in the dual gravitational counterpart of the non commutative $ \mathcal{N}=4 $ SYM plasma. In Section 3, we explicitly compute the holographic charge diffusion rates both along the commutative as well as the non commutative directions of the brane and found that unlike the case for the shear modes their ratio is different from the unity. Finally, we conclude in Section 4.

\section{The dual set up}
We start our analysis with a formal introduction to the geometrical construction in the bulk space time that is holographically dual to non commutative $ \mathcal{N}=4 $ SYM theory at strong coupling. It is already known from the earlier literature that non commutative gauge theories at strong coupling could be consistently obtained from string theory by considering the so called decoupling limit in a system of $ Dp $ branes in the presence of a background NS B field that gives rise to certain scale of non commutativity in the large $ N $ limit \cite{Seiberg:1999vs}-\cite{Hashimoto:1999ut}. To start with, we consider the non commutative $ \mathcal{N}=4 $ SYM theory at finite temperature whose dual counterpart in the string frame reads as \cite{Landsteiner:2007bd},
\begin{eqnarray}
ds_{10}^{2}&=&\mathcal{H}^{-1/2}(-f dt^{2}+dx^{2}+h(dy^{2}+dz^{2}))+\mathcal{H}^{1/2}(f^{-1}dr^{2}+r^{2}d\Omega^{2}_{5})\nonumber\\
f&=& 1- \frac{r_H^{4}}{r^{4}},~~h=\frac{1}{1+\Theta^{2}\mathcal{H}^{-1}},~~\mathcal{H}=\frac{L^{4}}{r^{4}}
\end{eqnarray}
where, $ \Theta $ is the so called non commutative parameter and $ r_H $ is the usual position of the horizon. Following the AdS/CFT prescription, one could write $ L^{4}=4 \pi g^{2}_{YM}N \alpha'^{2} $ which in the decoupling ($ \alpha' \rightarrow 0 $) limit corresponds to a large value of $ N $ where $ N $ is the number of $ D3 $ branes. Finally, setting $ u = r_H^{2}/r^{2} $ the effective five dimensional metric in the Einstein frame could be formally expressed as \cite{Landsteiner:2007bd},
\begin{eqnarray}
ds^{2}&=& h^{-1/4}\mathcal{H}^{-1/2}(-f dt^{2}+dx^{2}+h(dy^{2}+dz^{2}))+\frac{L^{2}h^{-1/4}}{4u^{2}f}du^{2}\nonumber\\
f(u)&=& 1-u^{2},~~h(u)=\frac{u^{2}}{u^{2}+a^{2}},~~\mathcal{H}(u)= \frac{u^{2}}{u_T^{2}},~~u_T = \frac{r_H^{2}}{L^{2}},~~a=\Theta ~ u_T .\label{E1}
\end{eqnarray}

 Eq (\ref{E1}) is in fact the starting point of our analysis. In the above mentioned coordinate system (\ref{E1}) the horizon is placed at $ u=1 $ and the boundary is located at $ u=0 $. One should take a note on the fact that here $ (t,x) $ are the usual commutative directions whereas on the other hand, the other two spatial coordinates $ (y,z) $ exhibit the non commutative nature \cite{Landsteiner:2007bd}.
From (\ref{E1}), it is in fact quite evident that due to presence of the non commutativity along two of the spatial directions of the brane, the full $ SO(3) $ symmetry of the boundary theory is reduced down to $ SO(2) $ leaving the rotational invariance only over the $ (y-z) $ plane of the brane. Finally, from (\ref{E1}) it is in fact quite trivial to note down the corresponding Hawking temperature which for the present case turns out to be,
\begin{eqnarray}
T = \frac{1}{\pi u_T L}.
\end{eqnarray}

\section{Charge diffusion}
Based on the original prescription \cite{Policastro:2002se}-\cite{Kovtun:2008kx} for evaluating retarded Green's function corresponding to $ U(1) $ currents ($ J_{\mu} $), the purpose of the present section is to first make a systematic analytic investigation of the DC conductivity ($ \sigma_{DC} $) both along the commutative as well as the non commutative directions of the brane and then compute the corresponding R- charge diffusion(s) ($ \mathfrak{D} $) using the so called Einstein relation, $ \mathfrak{D}=\sigma_{DC}/ \chi $, where $ \chi $ is the charge susceptibility and $ \sigma_{DC} $ is the DC electrical conductivity that could be formally expressed as \cite{Policastro:2002se}-\cite{Kovtun:2008kx},
\begin{eqnarray}
\sigma_{DC} &=& - \lim_{\mathfrak{w} \rightarrow 0}\frac{1}{\mathfrak{w}}Im\ \mathcal{G}^{R}_{ii}(\mathfrak{w} , \mathfrak{q}=0)\nonumber\\
\mathcal{G}^{R}_{ii}(\mathfrak{w} , \mathfrak{q}=0) &=& -i \int d\tau\ d\textbf{x}\ e^{i\mathfrak{w} \tau}\ \Delta (t)\ \langle[J_i (\textbf{x}), J_i (0)]\rangle. \label{E13}
\end{eqnarray}

In order to compute the above quantity in (\ref{E13}) and thereby the charge diffusion ($ \mathfrak{D} $), we essentially study the dynamics of vector $ U(1) $ perturbations over the fixed back ground of the anisotropic black brane (\ref{E1}) \cite{Policastro:2002se}-\cite{Kovtun:2008kx}. Dynamics of these vector perturbations are in general governed by the Maxwell's action namely,
\begin{eqnarray}
S_M = -\frac{1}{4g^{2}_{M}}\int d^{5}x \sqrt{-g}\mathcal{F}_{ab}\mathcal{F}^{ab}\label{E14}
\end{eqnarray}
where, $ g^{2}_{M} $ stands for the Maxwell coupling of the $ U(1) $ theory.

The basic physics behind our analysis rests on the fact that the infra red behavior of these $ U(1) $ fluctuations in the bulk is solely governed by the \textit{hydrodynamics} where the dispersion relation of the type $ \mathfrak{w}=-i \mathfrak{D}\mathfrak{q}^{2} $ appears naturally as a consequence of the pole appearing in the Laplace transformed version of the charge density in the complex $ \mathfrak{w} $ plane which could be interpreted as a natural consequence of the diffusion of conserved charges.  In our analysis, considering the so called hydrodynamic limit namely, $ q \ll T $ we study fluctuations of the type, $\mathcal{A}_{m}\sim e^{i q.x}\mathcal{A}_{m}(t,u) $ over the background of (\ref{E1}). These fluctuations by means of the equation of motion as well as the relevant boundary conditions finally yield the dispersion relation of the above form in the limit $ \mathfrak{q}\rightarrow 0 $.

\subsection{Charge susceptibility}
The purpose of the present section is to compute the charge susceptibility ($ \chi $) corresponding to non commutative $ \mathcal{N}=4 $ SYM plasma at strong coupling. In the AdS/CFT framework, the dual geometry corresponding to this non commutative plasma (at finite temperature) is essentially described by the five dimensional black hole solution (\ref{E1}) in the bulk space time.

In our computations, we strictly follow the methods proposed in \cite{Kovtun:2008kx}. The bottom line of our analysis is the following: In order to compute the susceptibility ($ \chi $), one needs to systematically solve the temporal gauge field ($ \mathcal{A}_{t} $) in the bulk consistent with the boundary condition at the horizon ($ u=1 $).

The Maxwell equation that directly follows from (\ref{E14}) could be formally expressed as,
\begin{eqnarray}
\frac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}\mathcal{F}^{\mu\nu})=0.\label{E15}
\end{eqnarray}

 The equation of motion corresponding to $ \mathcal{A}_{t} $ that readily follows from (\ref{E15}) could be formally expressed as,
\begin{eqnarray}
\mathcal{A}''_{t}+\frac{\partial_u(\sqrt{-g}g^{tt}g^{uu})}{\sqrt{-g}g^{tt}g^{uu}}\mathcal{A}'_{t}=0.\label{E16}
\end{eqnarray}
The corresponding solution turns out to be,
\begin{eqnarray}
\mathcal{A}_{t}(u)= \mathfrak{C}_{2}+\frac{4\mathfrak{C}_{1} \left(a^2+u^2\right)^{7/8} \left(7 u^2 \, _2F_1\left(1,\frac{3}{2};\frac{13}{8};-\frac{u^2}{a^2}\right)-5 a^2\right)}{15 a^2 u^{3/4}}.\label{E7}
\end{eqnarray}

The coefficient $\mathfrak{C}_{2}$ is uniquely determined by demanding the fact that $ \mathcal{A}_{t} $ must vanish at the horizon ($ u=1 $) which yields,
\begin{eqnarray}
\mathfrak{C}_{2}= \frac{4 \mathfrak{C}_{1} \left(a^2+1\right)^{7/8} \left(5 a^2-7 \, _2F_1\left(1,\frac{3}{2};\frac{13}{8};-\frac{1}{a^2}\right)\right)}{15 a^2}.
\end{eqnarray}
On the other hand, the chemical potential is given by,
\begin{eqnarray}
\mu = \mathcal{A}_{t}(u)|_{u\rightarrow \varepsilon}=\mathfrak{C}_{2} -\frac{4\mathfrak{C}_{1} a^{7/4} }{3 \varepsilon^{3/4}}+\mathcal{O}(\varepsilon^{5/4})\label{E9}
\end{eqnarray}
where $ |\varepsilon|\ll 1 $. Clearly the above quantity in (\ref{E9}) diverges in the limit $ \varepsilon \rightarrow 0 $. In order to have a finite chemical potential for the bondary theory, we thereby define the renormalized chemical potential as,
\begin{eqnarray}
\mu_{R}= \lim_{\varepsilon \rightarrow 0}\left(\mu +\frac{4 \varepsilon}{3} \frac{\partial \mu}{\partial \varepsilon} \right)= \mathfrak{C}_{2}.\label{E10}
\end{eqnarray}

Finally, using (\ref{E7}) the charge density could be readily obtained as \cite{Kovtun:2008kx},
\begin{eqnarray}
\varrho = \frac{\delta S_{M}}{\delta \mathcal{A}_{t}}|_{u =0}= \frac{2 u_T \mathfrak{C}_{1}}{g^{2}_{M}L}.\label{E22}
\end{eqnarray}

Using (\ref{E10}) and (\ref{E22}), the charge susceptibility finally turns out to be,
\begin{eqnarray}
\chi = \frac{\varrho}{\mu_{R}}=\frac{15 a^{2}u_T}{2 g^{2}_{M}L \left(a^2+1\right)^{7/8} \left(5 a^2-7 \, _2F_1\left(1,\frac{3}{2};\frac{13}{8};-\frac{1}{a^2}\right)\right)}\approx \frac{u_{T}}{2g^{2}_{M}L}.\label{E12}
\end{eqnarray}
Interestingly here we note that the charge susceptibility ($ \chi $) (almost) does not get corrected in the non commutative parameter ($ a $) upto fifth orders in the perturbation series.

Having done these computations on charge susceptibility ($ \chi $), our next task would be to compute DC conductivities along both the commutative as well as non commutative directions of the brane. We denote $ \sigma_{\perp} $ as the conductivity along the commutative direction of the brane and $ \sigma_{\parallel} $ as the conductivity along the non commutative direction of the brane. Our purpose is to make a systematic comparison between these two conductivities and compare our results with the already existing results in the context of anisotropy \cite{Rebhan:2011vd}.

\subsection{Conductivity I: $ \sigma_{\perp} $}
As a first part of our analysis, we compute the DC electrical conductivity along one of the commutative directions of the brane, namely the $ x $ direction.
We consider fluctuations of the form,
\begin{eqnarray}
\mathcal{A}_{m}(u,t)=L \int d\mathfrak{w}e^{-i \mathfrak{w}t}\mathcal{A}_{m}(u).\label{E24}
\end{eqnarray}

Considering $ m =x $ and substituting (\ref{E24}) into (\ref{E15}), we obtain
\begin{eqnarray}
\mathcal{A}_{x}'' +\frac{\partial_u (\sqrt{-g}g^{uu}g^{xx})}{\sqrt{-g}g^{uu}g^{xx}}\mathcal{A}_{x}' -\mathfrak{w}^{2}\frac{g^{tt}}{g^{uu}}\mathcal{A}_{x}=0.\label{E25}
\end{eqnarray}

In order to solve the above equation (\ref{E25}) in the so called low frequency regime, we chose the following ansatz, namely,
\begin{eqnarray}
\mathcal{A}_{x}=(1-u)^{\alpha} \Psi (u).\label{E26}
\end{eqnarray}

Considering the so called in going wave boundary condition \cite{Policastro:2002se}-\cite{Kovtun:2008kx}, our first task is to explore the above equation (\ref{E25}) in the near horizon limit of the brane namely, $ u \sim 1 $. This essentially enable us to determine the coefficient $ \alpha $ uniquely. Substituting (\ref{E26}) into (\ref{E25}) and considering the incoming wave boundary condition near the horizon of the black brane it is in fact quite trivial to show that,
\begin{eqnarray}
\alpha = - \frac{i \mathfrak{w}}{4 \pi u_T^{3/2} T}.
\end{eqnarray}

Our next task would be to substitute (\ref{E26}) in to (\ref{E25}) and solve $ \Psi (u) $ perturbatively in the frequency $ \mathfrak{w} $ near the boundary of the space time. This will finally enable us to compute the DC conductivity ($ \sigma_{\perp} $). In order to solve $ \Psi (u) $ perturbatively in the frequency ($ \mathfrak{w} $) we consider the following perturbative expansion namely,
\begin{eqnarray}
\Psi (u)= \Psi ^{(0)} + i(\mathfrak{w}/T)\Psi^{(1)}+\mathcal{O}(\mathfrak{w}^{2}/T^{2})
\end{eqnarray}
where each of the individual coefficients satisfies equation of motion of the following form,
\begin{eqnarray}
\Psi''^{(1)}+\frac{1}{2 \pi u_T^{3/2} (1-u)}\left[ \Psi'^{(0)}+\frac{1}{2(1-u)}\Psi^{(0)}\right] +\frac{\partial_u (\sqrt{-g}g^{uu}g^{xx})}{\sqrt{-g}g^{uu}g^{xx}}\left[\Psi'^{(1)}+\frac{1}{4 \pi u_T^{3/2} (1-u)}\Psi^{(0)} \right] &=&0\nonumber\\
\Psi''^{(0)}+\frac{\partial_u (\sqrt{-g}g^{uu}g^{xx})}{\sqrt{-g}g^{uu}g^{xx}}\Psi'^{(0)}&=&0.\label{E18}\nonumber\\
\end{eqnarray}

In the following we quote corresponding solutions one by one. Let us first consider the second equation in (\ref{E18}). The corresponding solution turns out to be,
\begin{eqnarray}
\Psi^{(0)}= \frac{4 \mathfrak{d}_{1}\mathfrak{Z}}{195 u^{3/4} \sqrt[8]{a^2+u^2}}+\mathfrak{d}_{2} \label{e19}
\end{eqnarray}
where,
\begin{eqnarray}
\mathfrak{Z}=  u^2 \sqrt[8]{\frac{u^2}{a^2}+1} \left(13 \left(3 a^2+7\right) F_1\left(\frac{5}{8};\frac{1}{8},1;\frac{13}{8};-\frac{u^2}{a^2},u^2\right)
-20 u^2 F_1\left(\frac{13}{8};\frac{1}{8},1;\frac{21}{8};-\frac{u^2}{a^2},u^2\right)\right)
\nonumber\\
-65 \left(a^2+u^2\right). \label{E30}
\end{eqnarray}
 In the above, we have expressed solution (\ref{e19}) in terms of Appell polynomials where the coefficients $ \mathfrak{d}_{1} $ and $ \mathfrak{d}_{2} $ are related to each other through the condition $ \Psi^{(0)}(1)=0 $. On top of it, one can also impose the asymptotic normalization condition which for the present case turns out to be, $ \Psi^{(0)}(0)=1/L $. These two conditions should in principle sufficient to determine these unknown coefficients uniquely. However, for the present purpose of our analysis it is sufficient to know the boundary behaviour of the gauge fields since we will be finally evaluating the entities near the boundary of the space time. Expanding (\ref{E30}) near the boundary ($ u \sim 0 $) of the space time we note,
\begin{eqnarray}
\Psi^{(0)} \approx \frac{1}{L}\left( 1 - \frac{4 a^{7/4}}{3 \varepsilon^{3/4}}\right)^{-1}  \left( 1 - \frac{4 a^{7/4}}{3 u^{3/4}}+\frac{\left(8 a^2+7\right) u^{5/4}}{10 \sqrt[8]{a^2}}\right)+\mathcal{O}(u^{13/4})\label{E20}
\end{eqnarray}
where the numerical prefactor guarantees a normalized mode at the boundary. Note that here $ \varepsilon $ is the UV cut off as mentioned earlier. At the end of our calculations we finally consider the $ \varepsilon \rightarrow 0 $ limit in order to extract the finite piece at the boundary.

In the subsequent analysis we drop all the terms starting with quadratic order in $ u $.  Since $ u $ ranges between zero and one therefore it is indeed quite logical to truncate solutions upto certain order in $ u $ and particularly consider those terms that contribute significantly near the boundary of the space time. Using (\ref{E20}), the solution corresponding to ($ \Psi^{(1)} $) finally turns out to be,
\begin{eqnarray}
\Psi^{(1)} \approx \frac{1}{L}\left( 1 - \frac{4}{3 \varepsilon^{3/4}}\right)^{-1} \left( 1 -\frac{4 }{3 u^{3/4}}+\frac{a^{7/4}  \sqrt[4]{u}}{3 \pi  u_T^{3/2}}+\frac{a^{7/4} u^{5/4}}{6 \pi  u_T^{3/2}}-\frac{u}{4 \pi  u_T^{3/2}}\right) + \mathcal{O}(u^{2}).\label{E21}
\end{eqnarray}

Using (\ref{E20}) and (\ref{E21}), the non trivial piece in the DC conductivity (along $ x $- direction) finally turns out to be,
\begin{eqnarray}
\sigma_{\perp}= \frac{u_T}{g^{2}_{M}L T}.\label{eq23}
\end{eqnarray}
Finally, from (\ref{E12}) and (\ref{eq23}) one can easily read of the corresponding charge diffusion coefficient as,
\begin{eqnarray}
\mathfrak{D}_{\bot}=\sigma_{\perp}/\chi \sim \frac{1}{T}\label{E23}
\end{eqnarray}
where we have ignored the over all numerical pre factor. The above result (\ref{E23}) also follows from simple dimensional arguments. For example, it is straightforward to notice from
(\ref{E12}) that $ [\chi]=1/L^{2} $ since the dimension of the Maxwell coupling in five dimensions goes as $ [g^{2}_{M}]=L $ \cite{Kovtun:2008kx}. On the other hand, following the same line of arguments we note $ [\sigma_{\perp}]=1/L^{2} $. Using these facts it could be readily seen that $ [\mathfrak{D}_{\bot}]=L $, where we have used the fact $ [T]=1/L $.

Eq.(\ref{eq23}) is in fact an important observation in itself. It reveals certain important fact that the DC conductivity ($ \sigma_{\perp} $) along the commutative direction of the brane does not get modified due to the presence of the non commutative parameter ($ \Theta $). The same line of argument also holds for the corresponding charge diffusion rate ($ \mathfrak{D}_{\bot} $).

\subsection{Conductivity II: $ \sigma_{\parallel} $}
For the sake of completeness as well as the clarity, our final task would be to compute the DC conductivity along one of the non commutative directions of the brane, say the $ y $ direction and make a systematic comparison of our results  with the result obtained in the previous section. To do that we first turn on fluctuations of the type,
\begin{eqnarray}
\mathcal{A}_{y}(u,t)=L \int d\mathfrak{w}e^{-i \mathfrak{w}t}\mathcal{A}_{y}(u)
\end{eqnarray}
which satisfy differential equation of the following form,
\begin{eqnarray}
\mathcal{A}_{y}'' +\frac{\partial_u (\sqrt{-g}g^{uu}g^{yy})}{\sqrt{-g}g^{uu}g^{yy}}\mathcal{A}_{y}' -\mathfrak{w}^{2}\frac{g^{tt}}{g^{uu}}\mathcal{A}_{y}=0.\label{e25}
\end{eqnarray}

To solve (\ref{e25}), we choose the following ansatz,
\begin{eqnarray}
A_{y}=(1-u)^{\beta}\Phi(u)
\end{eqnarray}
where the coefficient $ \beta $ could be readily obtained from the near horizon data namely,
\begin{eqnarray}
\beta = - \frac{i \mathfrak{w}}{4 \pi u_T^{3/2} T}.
\end{eqnarray}

Like in the previous case, the function $ \Phi(u) $ could be solved perturbatively in the frequency $ \mathfrak{w} $ which in the hydrodynamic limit ($\mathfrak{w}/T \ll 1 $) yields the following set of equations namely,
\begin{eqnarray}
\Phi''^{(1)}+\frac{1}{2 \pi u_T^{3/2} (1-u)}\left[ \Phi'^{(0)}+\frac{1}{2(1-u)}\Phi^{(0)}\right] +\frac{\partial_u (\sqrt{-g}g^{uu}g^{yy})}{\sqrt{-g}g^{uu}g^{yy}}\left[\Phi'^{(1)}+\frac{1}{4 \pi u_T^{3/2} (1-u)}\Phi^{(0)} \right] &=&0\nonumber\\
\Phi''^{(0)}+\frac{\partial_u (\sqrt{-g}g^{uu}g^{yy})}{\sqrt{-g}g^{uu}g^{yy}}\Phi'^{(0)}&=&0.\label{e28}\nonumber\\
\end{eqnarray}
The corresponding solutions turn out to be,
\begin{eqnarray}
\Phi^{(0)} &= & \frac{1}{L}\left[1+ \frac{4 u^{5/4} \sqrt[8]{\frac{u^2}{a^2}+1} F_1\left(\frac{5}{8};\frac{1}{8},1;\frac{13}{8};-\frac{u^2}{a^2},u^2\right)}{5 \sqrt[8]{a^2+u^2}}\right] \nonumber\\
& \approx &\frac{1}{L}\left[ 1+\frac{4  u^{5/4}}{5 \sqrt[8]{a^2}}\right]+\mathcal{O}(u^{13/4}) \nonumber\\
\Phi^{(1)} & \approx & \frac{1}{L}\left[ 1+\frac{4u^{5/4} }{5}-\frac{ u}{4 \pi  u_T^{3/2}} \right]+\mathcal{O}(u^{2}).\label{e29}
\end{eqnarray}

Using (\ref{e29}), the corresponding DC conductivity finally turns out to be,
\begin{eqnarray}
\sigma_{\parallel}=\frac{u_T}{g^{2}_{M}LT}(1-\Theta^{1/4}u_T^{1/4}).\label{e31}
\end{eqnarray}

The way one would like to interpret the above result (\ref{e31}) is essentially the following: Unlike the previous case, the the DC conductivity ($ \sigma_{\parallel} $) along the non commutative direction of the brane is modified due to the presence of the non commutative parameter, and most importantly, the non commutative effects essentially suppress the value of the conductivity from that of its usual value corresponding to the commutative case. The same arguments also hold for the corresponding charge diffusion ($ \mathfrak{D}_{\parallel} $).

 Finally, the ratio between the two charge diffusion rates turn out to be,
\begin{eqnarray}
\frac{\mathfrak{D}_{\parallel}}{\mathfrak{D}_{\perp}} =\frac{\sigma_{\parallel}}{\sigma_{\perp}}= 1-\Theta^{1/4}u_T^{1/4}.\label{e32}
\end{eqnarray}
Eq.(\ref{e32}) is the full non perturbative result in the non commutative parameter ($ \Theta $) and is consistent with the corresponding result in the commutative ($ \Theta\rightarrow 0 $) limit. The crucial observation that one should make at this stage is the fact that unlike the case for the shear viscosity to entropy ($ \eta/s $) ratio \cite{Landsteiner:2007bd}, the charge diffusion rates are rather different along different directions of the brane. In other words, the charge diffusion is sensitive to the intrinsic anisotropy of the plasma. Finally, before we conclude, it is important to emphasis that similar observations have also been made earlier in a different context of anisotropy where people had observed, $ \sigma_{anisotropy}\neq \sigma_{isotropy} $ \cite{Rebhan:2011vd}.

\section{Summary and final remarks}
Let us now summarize the key findings of our analysis. In our analysis, considering the so called hydrodynamic limit, we explore the charge transport phenomena in non commutative $ \mathcal{N}=4 $ SYM plasma at strong coupling. The motivation of our current analysis rests on the earlier results on shear viscosity to entropy ($ \eta/s $) ratio which was found to be universal despite of the intrinsic anisotropy of the $ Dp $ brane \cite{Landsteiner:2007bd}. In our analysis, however we observe that unlike the case for the $ \eta/s $ ratio, the charge diffusion rates are indeed different along two different directions of the brane. In particular, we observe that the holographic DC conductivity gets significantly modified (\textit{only}) along the non commutative directions of the brane and its value is in fact turns out to be lower compared to its commutative counterpart. Therefore we might conclude that from the point of view of the charge transport property, both the $ \theta $ deformed as well as the non commutative $ \mathcal{N}=4 $ SYM theories exhibit some sort of similarity whereas on the other hand, they differ quite significantly when we compare them with respect to their shear channels. Finally, it is noteworthy to mention that our result smoothly matches to that with the corresponding commutative result in the limit of vanishing $ \Theta $.
\\ \\\
{\bf {Acknowledgements :}}
 The author would like to acknowledge the financial support from CHEP, Indian Institute of Science, Bangalore.\\

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