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.

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.