# Definition:Inflationary Mapping

(Redirected from Definition:Progressing Mapping)

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## Definition

Let $\left({S, \preceq}\right)$ be an ordered set.

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

Then $\phi$ is **inflationary** iff:

- $\forall s \in S: s \preceq \phi \left({s}\right)$

## Subset Ordering

Let $C$ be a set of sets or a class of sets.

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

Then $f$ is **inflationary** if and only if:

- $\forall x: x \in C \implies x \subseteq f \left({x}\right)$

That is, if and only if for each $x \in C$, $x$ is a subset of $f \left({x}\right)$.

## Also known as

An **inflationary mapping** can also be called **progressive** or **progressing**.