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


Sources