Definition:Immediate Predecessor Element

Definition
Let $\left({S, \preceq}\right)$ be an ordered set.

Let $a, b \in S$.

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

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

That is:
 * $a \prec b$ and $\left({a \,.\,.\, b}\right) = \varnothing$

where $\left({a \,.\,.\, b}\right)$ denotes the open interval from $a$ to $b$.

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

Also known as
Some sources just refer to a predecessor (element).

However, compare this with the definition on this site for predecessor element.

Also see

 * Definition:Precede
 * Definition:Strictly Precede


 * Definition:Immediate Successor Element


 * Immediate Predecessor in Toset is Unique