A lattice $H$ is a partial order such that each pair of elements $a$ and $b$ has
- a least upper bound $u \in H : a \leq u \land b \leq u$ and
- a greatest lower bound $l \in H : l \leq a \land l \leq b$
Transitivity is implied in the following definitions, e.g. $\{ (A, B), (B, C) \}$ implies $(A, C)$.
$H_1 = \{ (A, B), (B, C), (C, D) \}$ is a lattice as the above conditions hold true for all pairs $\{ A, B, C, D \} \times \{ A, B, C, D \}$:
A → B → C → D
$H_2 = \{ (A, B), (A, C), (A, D), (B, E), (C, F), (D, E), (D, F) \}$ is not a lattice as there is no least upper bound for $E, F$:
E F
|\ /|
B D C
\|/
A
What about $H_3 = \{ (A, B), (A, C), (B, D), (B, E), (C, D), (C, E), (D, F), (E, F) \}$?
F
/ \
D E
| X |
B C
\ /
A
Is this a lattice? Is $A$ the infimum of $(D, E)$ and $F$ the supremum of $(B, C)$?
To elaborate on the comment: $D\wedge E$ does not currently exist, but if you added restrictions like $D\leq E\leq B\leq C$ then you would have a lattice.