# 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

Let $C$ be a class.

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

Then $f$ is inflationary 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.

Some sources use progressing function.

## Also see

• Results about inflationary mappings can be found here.