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tawi
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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.