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Looking at the (very brief) definition in my textbook with no examples, I have the following:

A poset $(A,\preceq)$ in which every two elements have a greatest lower bound in $A$ and a least upper bound in $A$ is called a lattice.

But I can't think of a poset that doesn't have a GLB and LUB...

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3 Answers 3

up vote 10 down vote accepted

What about this one? What’s the least upper bound of the two fellows at the top?

        *   *  
         \ /  
          *
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$$\Huge\mathsf V$$ –  Asaf Karagila Mar 19 '13 at 0:33
1  
You could also mimic the infamous $\sf W$ directly, that would have worked equally well. –  Asaf Karagila Mar 19 '13 at 0:38
    
Thanks. Never considered this. This is why I need to study more :) –  agent154 Mar 19 '13 at 2:24
    
@agent154: You’re welcome. –  Brian M. Scott Mar 19 '13 at 2:24

I (and presumably William of Ockham) suggest a 2-element antichain.

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That was my original answer, but I replaced it with the sleek and dazzling succinct answer "$\ldots$" (a three-element antichain). –  Asaf Karagila Mar 19 '13 at 1:01

$$\Huge\ldots\vphantom{Some filler, if only there was a two-dots symbol}$$

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Antichains forever! –  Brian M. Scott Mar 19 '13 at 0:28
    
Brian, so antichains are Strawberry Fields? –  Asaf Karagila Mar 19 '13 at 0:32
    
Well, I think that you could safely say that Lennon was anti-chains! –  Brian M. Scott Mar 19 '13 at 1:31

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