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.
Hasse Digram Leveling
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discrete-mathematics
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0You'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. – 2012-11-11
2 Answers
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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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0@TMM: Yes, I saw that you had. I agree with you. – 2012-11-11
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$1$ 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.