A lattice is made out of pairs $(A, C)$ where $A$ can be $0,1,2,3$ and $C$ is a subset of $\{g,i,p\}$.
Why can't there be a path between $(1,\{g,p\})$ and $(2,\{p\})$? Isn't the defintion of $a\leq c$
$$(a_1,a_2,\ldots,a_n)\leq(c_1,c_2,\ldots,c_m)$$
if $(a_1,a_2\ldots,a_n)$ if there exists an index $i$ such as $0\leq i\leq n$ and $a_0=c_0$, $a_1=c_1$, $\ldots$, $a_i<c_i$ or if $m>n$ if all previous elements are equal. Wouldn't this apply here since $1<2$?
Thanks!