# Thread: Partial Order

1. ## Partial Order

Let the set S = {a, b, c, d, e}. Draw the diagram of a partial order with two
maximal elements on the set S.

My question is what does it mean to be a maximal element here?

2. Originally Posted by chiph588@
Let the set S = {a, b, c, d, e}. Draw the diagram of a partial order with two maximal elements on the set S.
My question is what does it mean to be a maximal element here?
A maximal element has no successor. A maximal element proceeds no other element.
In the diagram a & b are maximal.

3. but isn't c the successor of a & b?

and is that the answer? I'm sorry, I'm still confused on this subject.

4. if you have a partially ordered set $(S, \leq),$ then an element $s \in S$ is called a maximum element of $S,$ if $s \leq t$ implies that $s=t.$ for example, in your set, we can put this partial order:

$a \leq b, \ b \leq c, \ a \leq c, \ d \leq e.$ (of course $a \leq a, \ b \leq b, \ c \leq c, \ d \leq d, \ e \leq e$ is also a part of definition.) see that $\{c, \ e \}$ is the set of maximal elements of $S.$