Hasse Diagram confusion

• Mar 24th 2013, 05:15 PM
personofblah
Hasse Diagram confusion
So the homework problem I was doing is as follows:
Answer these questions for the partial order represented by this Hasse diagram.
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 {a, b, c}.
f ) Find the least upper bound of {a, b, c}, if it exists.
g) Find all lower bounds of {f, g, h}.
h) Find the greatest lower bound of {f, g, h}, if it exists.
Attachment 27685
I know the answers to the first four questions but I don't know what to do for the bounds stuff. I tried reading my textbook but it doesn't make much sense. Can someone please explain this to me? You don't need to necessarilly provide me with the specific answers but if you do please be clear :)
• Mar 25th 2013, 06:21 PM
johng
Re: Hasse Diagram confusion
In any poset (partially ordered set), an element u is an upper bound of a set S iff $s\leq u$ for all s in S. In particular for {a,b,c} both l and m are upper bounds from your diagram. j is not an upper bound since j is not greater than or equal to c. You can thus determines all upper bounds for {a,b,c}. Now the least upper bound of a set S is an upper bound v of S such that $v\leq u$ for any upper bound u of S. (By the anti symmetry property of $\leq$, if a least upper bound exists, it is unique.) Once you have listed all upper bounds, it is clear that k is the least upper bound. Similar analysis for lower bounds and greatest lower bound.