| Version 6 |
Version 5 |
| \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. |
\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. |
|
|
| \textbf{Remark}. If $w$ is infinite, then statement is not true. The proof of Dilworth's Theorem and its couterexample in the infinite case can be found in the reference below. |
\textbf{Remark}. If $w$ is infinite, then statement is not true. The proof of Dilworth's Theorem and its couterexample in the infinite case can be found in the reference below. |
|
|
| \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} |
