Definition:Immediate Predecessor Element

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

Let $a, b \in S$.

Then $a$ is the predecessor to $b$ iff:
 * $(1) \quad a \prec b$
 * $(2) \quad \neg \exists c \in S: a \prec c \prec b$

That is, there exists no element between $a$ and $b$ in the ordering.

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