# 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.