Definition:Immediate Successor Element

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

Let $a, b \in S$.

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

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

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

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

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

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

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

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

If $a$ immediately succeeds $b$, some sources will say that $a$ covers $b$.

Also see

 * Definition:Succeed
 * Definition:Strictly Succeed


 * Definition:Immediate Predecessor Element


 * Immediate Successor in Toset is Unique
 * Non-Maximal Element of Well-Ordered Class has Immediate Successor