# Definition:Inflationary Mapping

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## 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.