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Version 9 |
| \begin{thm}{Dilworth}. If $P$ is a poset with width $w<\infty$, then $w$ is also the smallest integer such that $P$ can be written as the union of $w$ chains. |
\textbf{Dilworth's Theorem}. If $P$ is a poset with width $w<\infty$, then $w$ is also the smallest integer such that $P$ can be written as the union of $w$ chains. |
| \end{thm} |
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| \textbf{Remark}. The smallest cardinal $c$ such that $P$ can be written as the union of $c$ chains is called the \emph{chain covering number} of $P$. So Dilworth's theorem says that if the width of $P$ is finite, then it is equal to the chain covering number of $P$. If $w$ is infinite, then statement is not true. The proof of Dilworth's theorem and its counterexample in the infinite case can be found in the reference below. |
\textbf{Remark}. The smallest cardinal $c$ such that $P$ can be written as the union of $c$ chains is called the \emph{chain covering number} of $P$. So Dilworth's theorem says that if the width of $P$ is finite, then it is equal to the chain covering number of $P$. If $w$ is infinite, then statement is not true. The proof of Dilworth's theorem and its counterexample in the infinite case can be found in the reference below. |
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| \begin{thebibliography}{6} |
\begin{thebibliography}{6} |
| \bibitem{jbn} J.B. Nation, ``Lattice Theory", \PMlinkexternal{http://www.math.hawaii.edu/~jb/lat1-6.pdf}{http://www.math.hawaii.edu/~jb/lat1-6.pdf} |
\bibitem{jbn} J.B. Nation, ``Lattice Theory", \PMlinkexternal{http://www.math.hawaii.edu/~jb/lat1-6.pdf}{http://www.math.hawaii.edu/~jb/lat1-6.pdf} |
| \end{thebibliography} |
\end{thebibliography} |
