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#### caffeinemachine

##### Well-known member
MHB Math Scholar

Wikipedia defines a 'graded Poset' as a poset $P$ such that there exists a function $\rho\to \mathbb N$ 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.

#### Evgeny.Makarov

##### Well-known member
MHB Math Scholar

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.
This is indeed a graded poset, but it is not a bounded poset. The latter has to have a least and a greatest elements.

MHB Math Scholar