Examples of Partially Ordered Sets I was if anyone could come up with some good examples of partially ordered sets with exactly $3$ maximal elements. For example: all proper subsets of $[3]$, the maximal elements are $\{1,2\},\{1,3\}\{2,3\}$. 
I'm having trouble of thinking of more though.
 A: A common example of a poset is the positive natural numbers with the relation of strict divisibility. That is, $xRy \iff x|y \wedge x \neq y$. You can use this relation to construct infinitely many posets with 3 maximal elements. For example, $\{1, 2, 3, 5\}$ or $\{1, 2, 3, 4, 5, 9, 25\}$ etc.
A: The silly example is a set with three disjoint inverted trees.  Another is your example with lots of other elements, all of which are less than only one of $1,2,3$
A:               *  *  *  
              |  |  |  
              *  *  *  
              |  |  |  
              *  *  *  
              |  |  |  
              *  *  *  
              |  |  |  
              *  *  *  
              |  |  |  
              .  .  .  
              .  .  .  
              .  .  .  

And you can fill in some diagonal lines, if you want, and even leave out some of the verticals, as long as every point below the top line has a route to the top:
              * * *  
              |\|/|  
              * * *  
              | | |  
              * * *  
              |/ /|  
              * * *  
              | | |  
              * * *  
              |/|\|  
              . . .  
              . . .  
              . . .  

