# 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 $b$**.

## Also see

- Immediate Successor under Total Ordering is Unique
- Non-Greatest Element of Well-Ordered Class has Immediate Successor

## Sources

- 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 14$: Order - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings: Exercise $14.19$ - 1967: Garrett Birkhoff:
*Lattice Theory*(3rd ed.): $\S\text I.3$ - 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 3$: Relations