orthomodular latticeOrthomodularLattice2007-01-10 12:55:432008-03-08 15:03:56Definitionorthomodular posetorthogonalorthogonal complementrelative orthogonal complement\usepackage{amssymb,amscd}
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\subsubsection*{Orthogonality Relations}
Let $L$ be an orthocomplemented lattice and $a,b\in L$. $a$ is said to be \emph{orthogonal} to $b$ if $a\le b^{\perp}$, denoted by $a\perp b$. If $a\le b^{\perp}$, then $b=b^{\perp\perp}\le a^{\perp}$, so $\perp$ is a symmetric relation on $L$. It is easy to see that, for any $a,b\in L$, $a\perp b$ implies $a\wedge b=0$, and $a\perp a^{\perp}$.
For any $a\in L$, define $M(a):=\lbrace c\in L\mid c\perp a\mbox{ and }1=c \vee a\rbrace$. An element of $M(a)$ is called an \emph{orthogonal complement} of $a$. We have $a^{\perp}\in M(a)$, and any orthogonal complement of $a$ is a complement of $a$.
If we replace the $1$ in $M(a)$ by an arbitrary element $b\ge a$, then we have the set $$M(a,b):=\lbrace c\in L\mid c\perp a\mbox{ and }b=c \vee a\rbrace.$$
An element of $M(a,b)$ is called an \emph{orthogonal complement} of $a$ \emph{relative to} $b$. Clearly, $M(a)=M(a,1)$. Also, for $a,c\le b$, $c\in M(a,b)$ iff $a\in M(c,b)$. As a result, we can define a symmetric binary operator $\oplus$ on $[0,b]$, given by $b=a\oplus c$ iff $c\in M(a,b)$. Note that $b=b\oplus 0$.
Before the main definition, we define one more operation: $b-a:=b\wedge a^{\perp}$. Some properties: (1) $a-a=0$, $a-0=a$, $0-a=0$, $a-1=0$, and $1-a=a^{\perp}$; (2) $b-a=a^{\perp}-b^{\perp}$; and (3) if $a\le b$, then $a\perp (b-a)$ and $a\oplus (b-a)\le b$.
\subsubsection*{Definition}
A lattice $L$ is called an \emph{orthomodular lattice} if
\begin{enumerate}
\item $L$ is orthocomplemented, and
\item (\emph{orthomodular law}) if $x\le y$, then $y=x\oplus (y-x)$.
\end{enumerate}
The orthomodular law can be restated as follows: if $x\le y$, then
$y=x\vee (y\wedge x^{\perp})$. Equivalently, $x\le y$ implies $y=(y\wedge x)\vee (y\wedge x^{\perp})$. Note that the equation is automatically true in an arbitrary distributive lattice, even without the assumption that $x\le y$.
For example, the lattice $\mathbb{C}(H)$ of closed subspaces of a hilbert space $H$ is orthomodular. $\mathbb{C}(H)$ is modular iff $H$ is finite dimensional. In addition, if we give the set $\mathbb{P}(H)$ of (bounded) projection operators on $H$ an ordering structure by defining $P\le Q$ iff $P(H)\le Q(H)$, then $\mathbb{P}(H)$ is lattice isomorphic to $\mathbb{C}(H)$, and hence orthomodular.
A simple example of an orthocomplemented lattice that is not orthomodular is the benzene:
\begin{equation*}
\xymatrix {
& 1 \ar@{-}[ld] \ar@{-}[rd] & \\
b \ar@{-}[d] & & a^{\perp} \ar@{-}[d] \\
a \ar@{-}[rd] & & b^{\perp} \ar@{-}[ld] \\
& 0 & }
\end{equation*}
Note that $a\le b$, but $a\vee (b\wedge a^{\perp})=a\vee 0=a\ne b$.
An nice example of an orthomodular lattice that is not modular can be found in the reference below.
\textbf{Remarks}.
\begin{itemize}
\item Orthomodular lattices were first studied by John von Neumann and Garett Birkhoff, when they were trying to develop the \PMlinkname{logic of quantum mechanics}{QuantumLogic} by studying the structure of the lattice $\mathbb{P}(H)$ of projection operators on a Hilbert space $H$. However, the term was coined by Irving Kaplansky, when it was realized that $\mathbb{P}(H)$, while orthocomplemented, is not modular. Rather, it satisfies a variant of the modular law as indicated above.
\item More generally, an \emph{orthomodular poset} $P$ is an orthocomplemented poset such that
\begin{enumerate}
\item given any pair of orthogonal elements $x,y\in P$ ($x\le y^{\perp}$), their greatest lower bound exists ($x\vee y$ exists). Simply put, $x\perp y$ implies $x\vee y\in P$.
\item for any $x,y\in P$ such that $x\le y$, the orthomodular law holds (the right hand side of the orthomodular law exists via the first condition).
\end{enumerate}
From this definition, we see that an orthomodular lattice is just an orthomodular poset that is also a lattice.
\end{itemize}
\begin{thebibliography}{8}
\bibitem{lb} L. Beran, {\em Orthomodular Lattices, Algebraic Approach}, Mathematics and Its Applications (East European Series), D. Reidel Publishing Company, Dordrecht, Holland (1985).
\end{thebibliography}