| Version 20 |
Version 19 |
| 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} |
|
|
| \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. |
|
|
| An element in a bounded lattice is \emph{complemented} if it has a complement. A \emph{complemented lattice} is a bounded lattice in which every element is complemented. |
An element in a bounded lattice is \emph{complemented} if it has a complement. A \emph{complemented lattice} is a bounded lattice in which every element is complemented. |
|
|
|
|
| \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. |
|
|
| \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*} |
|
|
| 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. |
|
\item A complemented lattice such that every element has a unique complement is said to be \emph{uniquely complemented}. |
|
|
|
Denote $a'$ the unique complement of $a$ in a uniquely complemented lattice. Then $$a''=a.$$ To see this, we have that $a\vee a'=1$, $a\wedge a'=0$, as well as $a''\vee a'=1$, $a''\wedge a'=0$. So $a=a''$, since they are both complements of $a'$. |
| \item If a complemented lattice $L$ is a distributive lattice, then $L$ is uniquely complemented (in fact, a Boolean lattice). For if $y_1$ and $y_2$ are two complements of $x$, then $$y_2=1\land y_2=(x\lor y_1)\land y_2= |
\item If a complemented lattice $L$ is a distributive lattice, then $L$ is uniquely complemented (in fact, a Boolean lattice). For if $y_1$ and $y_2$ are two complements of $x$, then $$y_2=1\land y_2=(x\lor y_1)\land y_2= |
| (x\land y_2)\lor(y_1\land y_2)=0\lor(y_1\land y_2)=y_1\land y_2.$$ Similarly, $y_1=y_2\land y_1$. So $y_2=y_1$. |
(x\land y_2)\lor(y_1\land y_2)=0\lor(y_1\land y_2)=y_1\land y_2.$$ Similarly, $y_1=y_2\land y_1$. So $y_2=y_1$. |
| \item In the category of complemented lattices, a morphism between two objects is a $\lbrace 0,1\rbrace$-lattice homomorphism; that is, a lattice homomorphism which preserves $0$ and $1$. |
\item In the category of complemented lattices, a morphism between two objects is a $\lbrace 0,1\rbrace$-lattice homomorphism; that is, a lattice homomorphism which preserves $0$ and $1$. |
| \end{itemize} |
\end{itemize} |
