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So, while I understand the difference between maximal elements and greatest elements, I'm having trouble understanding how to identify them on a lattice diagram (just as an example, the lattice diagram of the power set of A = {1,2,3}).

From what I understand (and this could be wrong), the element {1,2,3} of the power set would be both the greatest element (since {1,2,3} is drawn at least as high as every other element in the diagram) and maximal element (since nothing is drawn higher than {1,2,3}).

Any help will be greatly appreciated! Thanks.

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You might want to distinguish between Hasse diagrams (which work for all finite posets), and lattice diagrams (which only work for lattices) a little more here. Considering Austin Mohr's example, it's not too hard to draw a Hasse diagram for the power set of {1, 2, 3} without {1, 2, 3}. But, such a set is not a lattice, since {1, 2} and {2, 3} don't have a supremum. –  Doug Spoonwood Oct 23 '11 at 23:21
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Your example is correct.

To see when these two notions might be different, consider your Hasse diagram, but with the greatest element, $\{1,2,3\}$ , removed.

This diagram has no greatest element, since there is no single element above all other elements in the diagram.

The diagram has three maximal elements, namely $\{1,2\}$ , $\{1,3\}$ , and $\{2,3\}$ . Each of these is maximal because there is no element above them.

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