- #1
caffeinemachine
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Graded poset on wiki: Graded poset - Wikipedia, the free encyclopedia
Wikipedia defines a 'graded Poset' as a poset $P$ such that there exists a function $\rho:P\to \mathbb N$ such that $x< y\Rightarrow \rho(x)< \rho(y)$ and $\rho(b)=\rho(a)+1$ whenever $b$ covers $a$.
Then if you go to the 'Alternative Characterizations' on the page whose link I gave above you would see that the first line reads:
A bounded poset admits a grading if and only if all maximal chains in $P$ have the same length.
Here's the problem. Consider $P=\{a,b,c,d\}$ with $a<b,b<d,a<c,a<d$. All other pairs are incomparable. Then according to the first definition $P$ is a graded poset while the second definition says otherwise.
Maybe I am committing a very silly mistake but just can't find it.
Please help.
Wikipedia defines a 'graded Poset' as a poset $P$ such that there exists a function $\rho:P\to \mathbb N$ such that $x< y\Rightarrow \rho(x)< \rho(y)$ and $\rho(b)=\rho(a)+1$ whenever $b$ covers $a$.
Then if you go to the 'Alternative Characterizations' on the page whose link I gave above you would see that the first line reads:
A bounded poset admits a grading if and only if all maximal chains in $P$ have the same length.
Here's the problem. Consider $P=\{a,b,c,d\}$ with $a<b,b<d,a<c,a<d$. All other pairs are incomparable. Then according to the first definition $P$ is a graded poset while the second definition says otherwise.
Maybe I am committing a very silly mistake but just can't find it.
Please help.
