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