Definition:Immediate Successor Element

Definition
Let $\left({S, \preceq}\right)$ be a poset.

Let $a, b \in S$.

Then $a$ is the immediate successor (element) to $b$ iff $b$ is the immediate predecessor (element) to $a$.

That is, iff:
 * $(1): \quad b \prec a$
 * $(2): \quad \nexists c \in S: b \prec c \prec a$

We say that $a$ immediately succeeds $b$.

Also see

 * Immediate predecessor element