\begin{definition}[Definition:Immediate Successor Element]
Let $\struct {S, \preceq}$ be an ordered set.
Let $a, b \in S$.
Then $a$ is an '''immediate successor (element)''' to $b$ {{iff}} $b$ is an immediate predecessor (element) to $a$.
That is, {{iff}}:
:$(1): \quad b \prec a$
:$(2): \quad \nexists c \in S: b \prec c \prec a$
That is, there exists no element strictly between $b$ and $a$ in the ordering $\preceq$.
That is:
:$a \prec b$ and $\openint a b = \O$
where $\openint a b$ denotes the open interval from $a$ to $b$.
We say that '''$a$ ''immediately'' succeeds $b$'''.
\end{definition}