How to spot a lattice

Question

Let $A=\{1,2,3,4,5,6\}$ and let's consider the usual order relation given by $\leq$. My textbook includes an image representing this structure:

Given the definition of lattice, I don't understand how this isn't it. When I am looking for the supremum and the infimum for every pair it is obvious that $\forall (a,b)\in A$

$$\begin{aligned} \;a\leq6 \text{ and } b\leq6, 1\leq a \text{ and } 1\leq b\end{aligned}$$

so what is that I don't take into account?

Answer 1

The problem is that $2$ and $3$ have $2$ "smallest possible upperbounds" but they are not comparable so they don't have a join. Similarly, $4$ and $5$ don't have a meet.

Hope that helps,

Answer 2

Remember for example that $4\wedge 5$ will be the unique greatest lower bound for 4 and 5. Now 2 and 3 are both lower bounds, so $4\wedge 5 $ above those.. but how could that be possible? There isn't such a node...