Definition:Immediate Successor Element/Class Theory
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Definition
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
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text I$ -- Superinduction and Well Ordering: $\S 1$ Introduction to well ordering: Discussion