# Definition:Inflationary Mapping

## 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.