Consider the divisibility relation on the set

$$S = \{ 2, 6, 7, 14, 15, 30, 70, 105, 210 \}$$

It is given that this relation is a partial order on $S$.

(i) Draw the Hasse diagram for this partial order.

(ii) Find all maximal elements and all minimal elements of S.

(iii) Does $S$ have a greatest element? Does $S$ have a least element? If so, write them down; if not, explain why not.

I'm not sure if this is a reasonable problem to seek review for from math.stackexchange, but I perhaps I can at least check my solutions for (ii) and (iii).

For (i), my Hasse diagram is a mess: there are lines criss-crossing through other lines. I'm not sure if this is allowed, but if not, I'm not sure how else it can be done?

For (ii), I got that the maximal elements of $S$ are $\{ 210 \}$, since a maximal element of a subset $S$ of some partially ordered set (poset) is an element of $S$ that is not smaller than any other element in $S$, and the minimal elements of $S$ are $\{ 2, 7, 15 \}$, since a minimal element of a subset $S$ of some partially ordered set is defined as an element of $S$ that is not greater than any other element in $S$.

For (iii), I got that $S$ has a greatest element $\{ 210 \}$, since the greatest element of a subset $S$ of a partially ordered set (poset) is an element of $S$ that is greater than every other element of $S$; for the least element, I wrote that $S$ does not have a least element, since there is no element of $S$ that is less than every other element of $S$.

I would greatly appreciate it if people could please take the time to review this.

  • 2
    $\begingroup$ your answers are correct. In the (iii) part, you mistyped that $S$ has no minimal elements, It has minimal elements as you have already found them but no least element. $\endgroup$
    – Anurag A
    Aug 26 '18 at 19:29
  • $\begingroup$ @AnuragA thank your for the confirmation. I will fix my typo now. $\endgroup$ Aug 26 '18 at 19:38

Here is your Hasse diagram for the divisibility relation on $S$.

enter image description here

  • $\begingroup$ Ahh, thank you for this! Evidently, I made a mistake which made mine look horrible. Thanks for clearing that up! $\endgroup$ Aug 26 '18 at 19:46
  • $\begingroup$ I will point out that $15$ is a minimal element as well. Depending on what you prefer to prioritize, your Hasse diagram can look differently. Some people prefer to have all minimal elements appearing level with one another at the bottom, while others prefer to avoid as many crossings as possible. I will also point out that you are missing an edge: $105 = 3\times 5 \times 7$ so there should have been an edge from $7$ to $105$ heading up in the diagram as well. You can draw this diagram as a planar graph if you wish, but crossings seem unavoidable if all minimal are level at bottom. $\endgroup$
    – JMoravitz
    Aug 26 '18 at 19:49
  • $\begingroup$ @JMoravitz Thanks for pointing out the missing edge. As far as $15$ being minimal is concerned OP already had figured that out (see his post). $\endgroup$
    – Anurag A
    Aug 26 '18 at 19:59
  • $\begingroup$ @AnuragA Just to be clear: It is ok to have crossing lines in Hasse diagrams, right? Sometimes this is necessary? $\endgroup$ Aug 26 '18 at 23:27
  • $\begingroup$ @ThePointer Yes crossing lines are okay. $\endgroup$
    – Anurag A
    Aug 28 '18 at 20:16

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