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?