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I am having some trouble understanding lattices. The following is defined as not being a lattice:

$$S = \{ \{\}, \{y\}, \{x\}, \{x,y,z\}, \{a,x,y,z\}, \{b,x,y,z\} \}$$

I understand that a lattice is a poset for each exists a unique largest element and a unique smallest element (least upper bound and greatest lower bound).

Therefore, is the above not a lattice because we have: $\{a,x,y,z\}$ and $\{b,x,y,z\}$?

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Is the order at hand $\subseteq$? –  Git Gud Feb 10 '13 at 16:57
    
Yes, sorry. ⊆ is the order. –  user1796218 Feb 10 '13 at 17:03
    
You wrote "I understand that a lattice is a poset for each exists a unique largest element and a unique smallest element (least upper bound and greatest lower bound)". The part outside of the brackets doesn't really make sense. A lattice is a poset in which $\sup (\alpha ,\beta)$ exists for any elements $\alpha ,\beta$ in the lattice. –  Git Gud Feb 10 '13 at 17:27

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In a lattice, for any $\alpha ,\beta$, $\sup (\alpha ,\beta )$ exists.

If your set were to be a lattice, what would be $\sup (\{a,x,y,z\},\{b,x,y,z\})$?

If $\sup (\{a,x,y,z\},\{b,x,y,z\})$ existed in $S$, then there would exist upperbounds for $\{\{a,x,y,z\},\{b,x,y,z\}\}$, but none exist. Therefore $\sup (\{a,x,y,z\},\{b,x,y,z\})$ doesn't exist. Hence $(S,\subseteq )$ is not a lattice.

It is true that $\varnothing$ is the greatest lower bound for $S$ because it is the only lower bound for $S$.

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Sorry, I don't understand... –  user1796218 Feb 10 '13 at 17:03
    
@user1796218 I added some stuff to the answer. Basically I'm proving you're right about what you said in your question. –  Git Gud Feb 10 '13 at 17:07
    
Thank you! Stupid question: what is sup()? –  user1796218 Feb 10 '13 at 17:20
    
@user1796218 $\sup (\alpha, \beta)$ is the least upperbound for the set $\{\alpha ,\beta\}$. –  Git Gud Feb 10 '13 at 17:23
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@user1796218 There are two reasons for $\sup$ to not exist. Either there isn't any upper bound or there isn't a least upper bound. In our case there is no upper bound at all. Your argument has the right idea, but you're confused about somethings. For instance $\{a,x,y,z\}$ can't be the "largest element" of $S$ because it isn't bigger than $\{b,x,y,z\}$. –  Git Gud Feb 10 '13 at 18:32

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