# Hasse Digram Leveling

I have the following Hasse diagram for the divides relation on the set $\{1,2,3,4,5,6,7,8,9,10\}$. I felt that everything is in the right place, but my professor is saying that one of the elements is not on the correct level. I looked up some more information on how the levels work and it seems like the only condition is that if $x\preccurlyeq y$ then $x$ should be lower than $y$ in the Hasse diagram. With this in mind, I think that I can put $10$ on the same level as the $4,6,$ and $9$, but it doesn't seem incorrect to be where it is. Here is the definition on proofwiki.

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$2$ and $6$ should also be connected as $2|6$. – Julian Kuelshammer Nov 11 '12 at 19:35
You're right. I was doing the diagram on computer and missed that, but there was a line on the homework and so that is not the problem. – russjohnson09 Nov 11 '12 at 19:38

It’s traditional to place an element as close to the bottom of the Hasse diagram as possible; if that convention is followed, then $10$ does indeed belong on the same level as $4,6$, and $9$, and I’d bet that that’s what your instructor has in mind.

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Okay. I wasn't sure about that. I also added the line by the way. – russjohnson09 Nov 11 '12 at 19:41
@GEdgar: I consider this edit unnecessary and don’t approve of it; I’m letting it stand only because it was proposed by the OP. – Brian M. Scott Nov 11 '12 at 20:22
@Brian: I came past this suggested edit on the review and rejected it, so I'm surprised it was approved. In my opinion people should not add their own stuff to someone else's answer. – TMM Nov 11 '12 at 20:25
@TMM: Yes, I saw that you had. I agree with you. – Brian M. Scott Nov 11 '12 at 20:28

$1$ has $0$ prime factors.

Each of $2$, $3$, $5$, $7$ has $1$ prime factor.

Each of $4$, $6$, $10$ has $2$ prime factors.

$8$ has $3$ prime factors.

That doesn't make the Hasse diagram incorrect simply as a Hasse diagram, but for some purposes it's not as good.

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