# Definition:Immediate Predecessor Element

## Definition

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

Let $a, b \in S$.

Then $a$ is an **immediate predecessor (element)** to $b$ if and only if:

- $(1): \quad a \prec b$
- $(2): \quad \neg \exists c \in S: a \prec c \prec b$

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 $b$**.

### 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 predecessor (element)** to $b$ if and only if:

- $(1): \quad a \prec b$
- $(2): \quad \neg \exists c \in S: a \prec c \prec b$

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

## Also defined as

Some sources define an **immediate predecessor 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 predecessor (element)** as a **predecessor (element)**.

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

## Also see

## 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 $14.19$ - 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 3$: Relations