Definition:Immediate Predecessor 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 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