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Can someone help me to solve this problem.

Are these Hasse diagrams lattices?

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What are your thoughts? What have you tried? –  Clive Newstead Jan 4 '13 at 22:08
    
I know lattice:"A partially ordered set in which every pair of elements has both a least upper bound and a greatest lower bound is called a lattice"and i think first diagram is lattice. –  Miranda Miri Jan 4 '13 at 22:10
    
Okay $-$ and you're right. What about the others? –  Clive Newstead Jan 4 '13 at 22:11
    
But I need a tutorial or complete explain about lattice. –  Miranda Miri Jan 4 '13 at 22:18
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Correct! And the third? Don't worry, I'll write a more comprehensive response soon $-$ but it's worth properly thinking about them first. –  Clive Newstead Jan 4 '13 at 22:26
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(Following on the comments.) They're all latices. When you have a Hasse diagram, it's fairly easy to find greatest lower bounds and least lower bounds.

For instance, given $x,y$, if $x \le y$ then $x \vee y = y$ and $x \wedge y = x$. This is easy to spot because you can connect $x$ to $y$ by a path that moves in just one direction. In your first diagram, for example, you know that $a \le e$ because there's an upwards-directed path $a \to b \to e$.

Sometimes we don't have $x \le y$ or $y \le x$, e.g. $f$ and $g$ in your second diagram. To get from $f$ to $g$ you have to move up and down again, or down and up again, so they're not comparable. But that's still not a problem: just by looking at the diagram you can see that $f \vee g = h$ and $f \wedge g = b$.

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Thanks.Can you give me a Hasse diagram that is not lattice? –  Miranda Miri Jan 4 '13 at 22:53
    
Sure: $\vee$ is one $\wedge$ is another, and $\ \cdot\ \cdot\ $ is another. –  Clive Newstead Jan 4 '13 at 22:55
    
What is "\vee ", " \wedge", "\ \cdot\ \cdot\ "? –  Miranda Miri Jan 4 '13 at 22:59
    
Ah, the fact that you're writing that makes me think the MathJax rendering isn't working. "\vee" is meant to look like V, "\wedge" like Λ and "\ \cdot\ \cdot\ " like · · –  Clive Newstead Jan 4 '13 at 23:03
    
...it was my way of trying to draw Hasse diagrams. –  Clive Newstead Jan 4 '13 at 23:07
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2nd one is not a lattice. (b,c) have three Upper Bounds none of which is the least upper bound. The other two are lattices.

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