I am currently reading Proofs: A Long Form Mathematics Textbook written by Jay Cummings, and am on the section about Partial Orders. When discussing how to visualize a POSET he defines a Hasse diagram as follows:
Also note I believe the author intended to say A = powerset({1,2,3})
I am confused as if you look at the diagram he has drawn it doesn't match the definition he gives. The sets {1,3} and {2} are incomparable with inclusion so according to his rules should be draw on the same vertical level, but are drawn on different vertical levels in the diagram. Same thing with {1,2} and {3}.
When I looked online every definition of Hasse diagrams I found didn't make any mention to drawing incomparable elements on the same vertical level. They only mentioned how if an element a covers an element b(meaning a≤b and there is no c for which a≤c<b) then a is placed below b, and a line is draw from a to b.
My current guess is that the author included an incorrect definition of a Hasse diagram. Is there something I am missing?