# Discrete Math - posets

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1. Nov 24, 2015

### tawi

1. The problem statement, all variables and given/known data
My task is to find out what is the lowest # of elements a poset can have with the following characteristics. If such a set exists I should show it and if it doesn't I must prove it.

1) has infimum of all its subsets, but there is a subset with no supremum
2) has two maximal and two minimaln elements
3) has two greatest elements
4) has one minimal but no least element

2. Relevant equations

3. The attempt at a solution

2) should be easy. We can just take Hasse diagram for divides relation of the set
{3,5} and we get two maximal and two minimal elements.

3) should be impossible since greatest/least element can only be one.

4) seems like it should be impossible (at least in fininte sets) as well even though I am not sure on this one

1) Again, if we take divides relation on the set {1,2,3} then 1 is the lower bound of all the subsets.
On the other hand the subset {2,3} does not have upper bound because the least upper bound is 6. But 6 is not in our original set.

Does that seem alright and is 3 the least number of elements a set satisfying this can have?
And what about the other way around? Is there a set that has upper bounds of all its subsets but there is a subset with no lower bound?

Thanks for any help.

2. Nov 24, 2015

### Staff: Mentor

Could you explain what a poset is and which kind of ordering you assume? Of course can there be more than one "greatest" element.
E.g. 2ℤ ⊃ 4ℤ ⊃ 8ℤ ⊃ 16ℤ ⊃ ... and 3ℤ ⊃ 6ℤ ⊃ 12ℤ ⊃ 36ℤ ⊃ ... and both 2ℤ and 3ℤ are maximal.

3. Nov 24, 2015

### tawi

Well "poset" as partially ordered set. What kind of ordering am I assuming? That is up to me, my task is to find out whether such a set exists and if so, what is the least amount of elements it can have.

4. Nov 24, 2015

### Staff: Mentor

Let's light another candle: What's a great element and how it differs from a maximal element? Same with least and minimal? I assume infimum and supremum are related to the partial order ⊆ since you use it in connection with subsets. But if so, then your general set you started with is always a supremum and the empty set an infimum. But then you require a subset without supremum but a maximal element. I'm confused.

5. Nov 25, 2015

### haruspex

I agree with your answers on 1, 2, and 3. Can you prove 3 is the least number in q1?
I believe 4 is possible with a countably infinite set. Try to prove it is not possible with a finite set.