Does anyone know how to identify the minimal and maximal elements of the following POSET other than visually using a Hasse diagram?
For example, the Hasse diagram from wiki shows the set P of divisors of 60, partially ordered by the relation "x divides y". The red subset $S = \{1,2,3,4\}$ has two maximal elements, viz. 3 and 4, and one minimal element, viz. 1, which is also its least element.
I'm trying to evaluate this using the Cartesian product of $(P, P_/)$. For example with the subset filled in. Is this the correct way to start analyzing such a problem?
4 x x x
3 x x
2 x x
1 x
1 2 3 4 5 6 12 10 15 20 30 60
My first thoughts are to visit each row in the $Y$ axis. The row with the most positive relations is a maximum. If the row is false for any binary pair, then there may be another maximum. The next row is checked and if it is positive for the false pair, then it is another maximum. I'm hoping there is a better mathematical solution to establish this?
I have to confess to not really understanding what it is you want out of this question. Are you literally only interested in the exact 12-element poset in the picture, and if so, what's wrong with just looking at the picture? The only issue where a picture might let you down is if you've drawn the picture wrong, which in this case, you haven't, so you're good to go. Of course you could write out all of the order relations in a different form to a Hasse diagram. You could write them in a table. You could write them in a list. You could write them out between curly brackets as a subset of $P\times P$. But you'll be doing the same thing, that is, to answer your question "how to identify the minimal and maximal elements" of your poset, you will be "checking by hand", that is, you ... just ... see which ones are maximal and minimal. There is no advantage to your proposed algorithm since it is doing exactly what you do when you look at the picture, that is, for a potential maximal element, checking there is no other element above it, and for a potential minimal element, checking there is no other element below it.
If you are looking for a method to find maximal and minimal elements under divisibility for any subset of the natural numbers, the really useful thing we can use (that is, to facilitate this "checking" described above) is unique factorisation into primes. If we write each number as a product of its prime factors then we can easily read off divisibility relations between numbers, since a number $m$ divides a number $n$ if and only if all the prime factors (with multiplicity) of $m$ are prime factors of $n$.