Question

Draw the Hasse diagrams of all partial ordered sets with at most 4 elements. Which of these are lattices?

Answer 1

R={(1,1), (1,2), (1,3), (1,4), (1,6), (1,8), (1,12), (2,2), (2,4), (2,6), (2,8), (2,12), (3,3), (3,6), (3,12), (4,4), (4,8), (4,12), (6,6), (6,12), (8,8), (12,12)}.

Step 1.You make a directed graph corresponding to a relation R.

Step 2. You discard all loops from the diagram and all transitive edges.

Step 3. We make sure that the initial vertex is below the terminal vertex and remove all arrows.

The minimal element is 1.

The maximal elements are 8 and 12.

The greatest element does not exist as there is no element that prospers all other elements.