I created a Hasse diagram for the poset $(\Box_4^2, \leq)$ with $\Box_4 = \{0, 1, 2, 3\}$ and $\leq$ being defined as
$$(a,b) \leq (c,d) \quad\Longleftrightarrow\quad (a<c \text{ and } b<d) \;\text{ or }\; (a=c \text{ and } b=d).$$
So the relation $\leq$ takes in $2$ pairs of numbers. Two pairs are in a relation if they're equal to each other or if both elements of the pair are smaller than the elements of the other pair.
The diagram looks like: Hasse diagram
Some elements have a relation with an element that is more than one level above them however, and I'm thinking this diagram might be faulty.
Edit: I have constructed a new diagram. I believe this poset does not have a least upper bound nor a greatest lower bound, is this correct?
