Definition:Immediate Successor Element

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Definition

Let $\left({S, \preceq}\right)$ be a poset.

Let $a, b \in S$.

Then $a$ is the immediate successor (element) to $b$ iff $b$ is the immediate predecessor (element) to $a$.

That is, iff:

$(1): \quad b \prec a$

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

That is, there exists no element strictly between $b$ and $a$ in the ordering.

We say that $a$ immediately succeeds $b$.

Some sources just refer to the successor (element).

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