\begin{theorem}
\label{thm:algebra}
Consider the algebra  $\la \mathbf{\cal 
T}_{\S},\emptyset,\bigcup,\mbc,\mna\ra$ under set union, patterns 
intersection and complementation. Then the following holds:
\ben
\item
$\cM\mbc\cM = \cM$.
\item
$\cM\mbc\cN = \cM\mba\cN$.
\item\label{it:DM}
$\cM\mbc(\cN\cup\cP)= (\cM\mbc\cN)\cup(\cM\mbc\cP)$.
\item
$\cM\mbc(\cN\mbc\cP) = (\cM\mbc\cN)\mbc\cP$.
\item
$\mna\mna \cM=\cM$.
\item
$\mna \cal T_{\S}= \emptyset$.
\item
$\mna \emptyset=\cal T_{\S}$.
\enen
\end{theorem}
\begin{proof}
\ben
\item
From idempotency of $\mcap$, Corollary \ref{co:unifpr}.
\item
From commutativity of $\mcap$, Corollary \ref{co:unifpr}.  
\item 
Immediate from Corollary \ref{co:setunif} and properties of 
set-theoretic union.
\item 
By induction on $\cP$.  If the latter is empty, the result is 
immediate.  Otherwise $\cM\mbc(\cN\mbc 
\{M\}\cup\cP)=\cM\mbc((\cN\mbc\{M\})\cup(\cN\mbc\cP)) = 
(\cM\mbc(\cN\mbc\{M\}))\cup)\cM\mbc(\cN\mbc\cP))$ by two application 
of (\ref{it:DM}).  Then, by IH, 
$(\cM\mbc\cN)\mbc\{M\}))\cup)(\cM\mbc\cN)\mbc\cP))$ and by 
(\ref{it:DM}) again conclude $(\cM\mbc\cN)\mbc \{M\}\cup\cP)$.  
\item
$\GG\vd M\in \|\mna\mna \cM\|$ iff by Corollary \ref{lemma:setpart} 
$\GG\not\in\|\mna \cM\|$ iff, again by Corollary \ref{lemma:setpart} 
$\GG\vd M\in \|\cM\|$.
\item
Immediate from Corollary \ref{co:setpart}.
\item
Immediate from Corollary \ref{co:setpart}.
\enen
\end{proof}

