Definition:Successor Mapping

Definition

Let $P$ be a set such that $P \ne \varnothing$.

Let $s: P \to P$ be a mapping such that:

$(1): \quad \forall m, n \in P: s \left({m}\right) = s \left({n}\right) \implies m = n$

$(2): \quad \operatorname{Im} \left({s}\right) \ne P$

$(3): \quad \forall A \subseteq P: \left({\exists x \in A: \neg \left({\exists y \in P: x = s \left({y}\right)}\right) \land \left({z \in A \implies s \left({z}\right) \in A}\right)}\right) \implies A = P$

These can be written in natural language as:

$(1): \quad s$ is injective.

$(2): \quad s$ is not surjective.

$(3): \quad $ For any subset $A$ of $P$ which has an element of $P$ which is the image of no element under $s$, such that it holds the image under $s$ of every number in it, is the same set as $P$.

The mapping $s: P \to P$, with the properties defined above, is known as the successor mapping or successor function.

The image element $s \left({x}\right)$ of an element $x$ is called the successor element or just successor of $x$.

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 16$