Let $A$ be a sequence $ a_0, a_2, ..., a_{n - 1} \in S$. Let $R \subseteq S \times S$ be a relation. I'm looking how to name those two properties:
$$ P_1(A, R) = \forall_{0 \le i < n-1} R(a_i, a_{i+1}) \\ P_2(A, R) = \forall_{0 \le i < j \le n-1} R(a_i, a_j) $$
I'd call first one monotonic and second one sorted but I'm not sure if those are correct names.
The first is the successor relation.
The second, using "<= n - 1" instead of "< n - 1",
is the strict transitive closure of the first.
With "i <= j" instead of "i < j", it is the transitive closure.