Discrete Math: Poset Characteristics and Minimum Element Count

[tawi]

Homework Statement

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

Homework Equations

The Attempt at a Solution

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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.

 

[fresh_42]

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.

 

[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.

 

[fresh_42]

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.

 

[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.