# Definition:Succeed

## Definition

Let $\left({S, \preceq}\right)$ be an ordered set.

Let $a, b \in S$ such that $a \preceq b$.

Then $b$ succeeds $a$.

## Also known as

The statement $b$ succeeds $a$ can be expressed as $b$ is a succcessor of $a$.

If it is important to make the distinction between a successor and a strict successor, the term weak successor can be used for successor.

When the underlying set $S$ of the ordered set $\left({S, \leqslant}\right)$ is one of the sets of numbers $\N$, $\Z$, $\Q$, $\R$ or a subset, the term is greater than or equal to is usually used instead of succeeds.

## Also defined as

Some sources use the term successor to mean immediate successor.