# Definition:Inflationary Mapping

## Definition

Let $\struct {S, \preceq}$ be an ordered set.

Let $\phi: S \to S$ be a mapping.

Then $\phi$ is **inflationary** if and only if:

- $\forall s \in S: s \preceq \map \phi s$

### Subset Ordering (Progressing Mapping)

Let $C$ be a class.

Let $f: C \to C$ be a mapping from $C$ to $C$.

Then $f$ is a **progressing mapping** if and only if:

- $x \in C \implies x \subseteq \map f x$

That is, if and only if for each $x \in C$, $x$ is a subset of $\map f x$.

## Also known as

An **inflationary mapping** is also known as a **progressive mapping** or **progressing mapping**, particularly in the context of class theory, where the ordering on the underlying ordered set is the subset relation.

The term **extensive mapping** can also occasionally be seen, but this is not endorsed on $\mathsf{Pr} \infty \mathsf{fWiki}$ because of its possible confusion with the concept of the Axiom of Extensionality.

Sources which prefer the term **function** to **mapping** will tend to use such here: **inflationary function**, **progressing function**, and so on.

## Also see

- Results about
**inflationary mappings**can be found**here**.