# What exactly is a lattice? And can somebody give an example of something that is not one?

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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        *   *
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$$\Huge\mathsf V$$ –  Asaf Karagila Mar 19 '13 at 0:33
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

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

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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

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