# Definition:Immediate Successor Element

## Definition

Let $\struct {S, \preceq}$ be an ordered set.

Let $a, b \in S$.

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

That is, if and only if:

$(1): \quad b \prec a$
$(2): \quad \nexists c \in S: b \prec c \prec a$

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 $b$.

### Class Theory

In the context of class theory, the definition follows the same lines:

Let $A$ be an ordered class under an ordering $\preccurlyeq$.

Let $a, b \in A$.

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

That is, if and only if:

$(1): \quad b \prec a$
$(2): \quad \nexists c \in S: b \prec c \prec a$

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

## Also defined as

Some sources define an immediate successor element only in the context of a total ordering.

However, the concept remains valid in the context of a general ordering.

## Also known as

Some sources just refer to an immediate successor (element) as 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 $b$.