Which of the following are Hasse diagrams of lattices?

Question

I'm trying to figure out why each of the following figures are not Hasse diagrams of lattices. Could someone, for example, explain why (A) is not a lattice? Thanks!

Answer 1

In (A) number the vertices from top to bottom and left to right, so that the top vertex is $v_1$, the three immediately below it are $v_2,v_3$, and $v_4$, and the bottom vertex is $v_{11}$. Show that vertices $v_5$ and $v_{10}$ have no join (least upper bound).

Numbering the vertices in (B) similarly, consider $v_{11}$ and $v_{13}$: $v_3$ and $v_5$ are minimal upper bounds, but $v_{11}$ and $v_{13}$ have no join — no upper bound that is less than or equal to all upper bounds.

See if you can manage (C) on your own now. I’ve left a bit of a hint in the spoiler-protected block below.

Two of the vertices in the bottom row of four will work.

Answer 2

The following theorem about partially ordered sets ("posets") is helpful.

Theorem: Suppose $P$ is a finite poset. Consider two elements $x,y \in P$ that don't have a join; so $x \vee y$ doesn't exist. Then either:

• $x$ and $y$ have no upper bound
• $x$ and $y$ have two or more (distinct) minimal upper bounds.

Therefore, my hint is:

Hint: To show that a finite poset isn't a lattice, look for pairs of elements that have two or more (distinct) minimal upper bounds.