| Version 13 |
Version 12 |
| Let $L$ be a bounded lattice (with $0$ and $1$), and $a\in L$. A \emph{complement} of $a$ is an element $b\in L$ such that |
Let $L$ be a bounded lattice (with $0$ and $1$), and $a\in L$. A \emph{complement} of $a$ is an element $b\in L$ such that |
| \begin{quote}$a\land b=0$ and $a\lor b=1$.\end{quote} |
\begin{quote}$a\land b=0$ and $a\lor b=1$.\end{quote} |
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| \textbf{Remark}. Complements may not exist. If $L$ is a non-trivial chain, then no element (other than $0$ and $1$) has a complement. This also shows that if $a$ is a complement of a non-trivial element $b$, then $a$ and $b$ form an antichain. |
\textbf{Remark}. Complements may not exist. If $L$ is a non-trivial chain, then no element (other than $0$ and $1$) has a complement. This also shows that if $a$ is a complement of a non-trivial element $b$, then $a$ and $b$ form an antichain. |
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| A \emph{complemented lattice} is a bounded lattice in which every element has a complement. |
A \emph{complemented lattice} is a bounded lattice in which every element has a complement. |
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| \textbf{Remarks}. |
\textbf{Remarks}. |
| \begin{itemize} |
\begin{itemize} |
| \item In a complemented lattice, there may be more than one complement corresponding to each element. Two elements are said to be \emph{related}, or \emph{perspective} if they have a common complement. For example, the following lattice is complemented. |
\item In a complemented lattice, there may be more than one complement corresponding to each element. Two elements are said to be \emph{related}, or \emph{perspective} if they have a common complement. For example, the following lattice is complemented. |
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| \begin{equation*} |
\begin{equation*} |
| \xymatrix{ |
\xymatrix{ |
| & 1 \ar@{-}[ld] \ar@{-}[d] \ar@{-}[rd] & \\ |
& 1 \ar@{-}[ld] \ar@{-}[d] \ar@{-}[rd] & \\ |
| a \ar@{-}[rd] & b \ar@{-}[d] & c \ar@{-}[ld] \\ |
a \ar@{-}[rd] & b \ar@{-}[d] & c \ar@{-}[ld] \\ |
| & 0 & |
& 0 & |
| } |
} |
| \end{equation*} |
\end{equation*} |
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| Note that none of the non-trivial elements have unique complements. Any two non-trivial elements are related via the third. |
Note that none of the non-trivial elements have unique complements. Any two non-trivial elements are related via the third. |
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| A complemented lattice such that every element has a unique complement is said to be \emph{uniquely complemented}. |
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| \item If a complemented lattice $L$ is a distributive lattice, then every element of $L$ has a unique complement. For if |
\item If a complemented lattice $L$ is a distributive lattice, then every element of $L$ has a unique complement. For if |
| $y$ and $y^{\prime}$ are two complements of $x$, then |
$y$ and $y^{\prime}$ are two complements of $x$, then |
| $$y^{\prime}=1\land y^{\prime}=(x\lor y)\land y^{\prime}= |
$$y^{\prime}=1\land y^{\prime}=(x\lor y)\land y^{\prime}= |
| (x\land y^{\prime})\lor(y\land y^{\prime})=0\lor(y\land |
(x\land y^{\prime})\lor(y\land y^{\prime})=0\lor(y\land |
| y^{\prime})=y\land y^{\prime}.$$ Similarly, $y=y^{\prime}\land y$. |
y^{\prime})=y\land y^{\prime}.$$ Similarly, $y=y^{\prime}\land y$. |
| So $y^{\prime}=y$. |
So $y^{\prime}=y$. |
| \item In a complemented distributive lattice $L$, denote $\overline{x}$ to be the \emph{unique} complement of $x$. It is not hard to see |
\item In a complemented distributive lattice $L$, denote $\overline{x}$ to be the \emph{unique} complement of $x$. It is not hard to see |
| that $\overline{\overline{x}}=x$. Next, for any $x,y\in L$, define $x+y$ to be |
that $\overline{\overline{x}}=x$. Next, for any $x,y\in L$, define $x+y$ to be |
| $(x\land\overline{y})\lor(\overline{x}\land y)$, the $L$ together |
$(x\land\overline{y})\lor(\overline{x}\land y)$, the $L$ together |
| with $\land$ and $+$ becomes a Boolean ring. |
with $\land$ and $+$ becomes a Boolean ring. |
| \end{itemize} |
\end{itemize} |
