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