For a given set $A$, such that $A = \{a, b, c, d\}$, and a relation on $A$, $R=\{(a,a),(b,b),(c,c),(d,d)\}$
Since transitivity is defined as: $$\forall x,y,z \in A : (x R y \land y R z)\to x R z$$
And if $x, y, z = a$ for instance, then we have all $(x,y), (y,z),(x,z)$ and so on for all other elements in $A$.
Does this mean that the relation satisfies all requirements for transitivity?
Yes. Such a relation is indeed a transitive relation, since the only relevant cases for the premise "$xRy \land yRz$" are $x=y=z$ in such relations. Since the premise never holds for cases where $x,y,z$ are not all the same, there is no need to consider them.
Mostly just posting this to get this out of the unanswered queue. Posting as Community Wiki in particular since I have nothing further to add.