The question I am looking at is, "Answer these questions for the poset $(\{3,5,9,15, 24,45\},|)$."
a) Find the maximal elements.
b)Find the minimal elements.
c) Is there a greatest element?
d) Is there a least element?
e) Find all upper bounds of $\{3,5\}$.
f) Find the least upper bound of$\{3,5\}$, if it exists.
g) Find all lower bounds of $\{15,45\}$.
h)Find the greatest lower bound of $\{15,45\}$, if it exists.
I have answers, I just want to make certain that I answered them properly, used the proper reasoning to answer them, and used the terminology correctly.
For a): For elements in the poset to be a maximal element, it has to be divisible by all the elements it is comparable to. The elements 24 and 45 are maximal, because they are divisible by every element they are comparable to. The reason why neither maximal element is comparable to the other is because $24|45$ and $45|24$ are both false statements.
For b): For elements in the poset to be minimal, is for them to be able to divide all other elements, in the relation, they are comparable to. $3|5$ and $5|3$ are both false statements, meaning they are incomparable elements, and since they divide every element they are comparable, they are minimal elements of the poset.
For c): No, because 24 and 45 are incomparable elements, meaning one doesn't precede the other.
For d): No, because 3 and 5 are incomparable elements, meaning one doesn't precede the other.
For e): To find the upper-bound of the set $\{3,5\}$, with respect to the poset, is to find all of the elements that both 3 and 5 divide. These elements are 15 and 35.
For f): To find the least upper-bound of the set $\{3,5\}$, is to consider the upper-bounds, and find the one that divides the others. $15|45$, so 15 is the least upper-bound.
For g): To find the lower bounds of the set $\{15,45\}$, with respect to the poset, is to find the eleents that divide into 15 and 45. These elements are 3,5, and 15.
For h): To find the greater lower bound of the set $\{15,45\}$, is to consider the lower bounds, and see which one is divisible by them all. $3|15$ and $5|15$, therefore 15 is the greatest lower bound.
I have a few other questions. If you have more than one extremal element, is it possible to have a greatest or least element? For parts a through d, I used this comparability argument; but now I don't feel so confident that it was the proper argument to use. If the comparability argument is correct, could someone explain to me why?--I answered this question yesterday, yesterday was so long ago.