\begin{definition}[Definition:Immediate Predecessor Element]
Let $\struct {S, \preceq}$ be an ordered set.
Let $a, b \in S$.
Then $a$ is an '''immediate predecessor (element)''' 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 strictly between $a$ and $b$ 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'' precedes $b$'''.
\end{definition}