---
tag: Z7P561Y2JQ
kind: concept
name: Interior operator
msc: [06A15]
sources:
  - url: https://en.wikipedia.org/wiki/Interior_algebra
---

## Summary

An interior operator on a [[partially ordered set|011C29Q1S1]] is a map that is deflationary, monotone and idempotent.
It is the dual of a [[closure operator|17FVAN6HXX]]: it sends each element to the greatest "open" element below it.
The [[interior|0C38S0KMJK]] of a subset of a topological space is the main example.
Interior operators are also called kernel operators.

## Explanations

### L1 human

An interior operator shrinks an element to the largest element of a special kind inside it.
In the plane, the interior of a closed disc is the open disc: remove the boundary circle.
Taking the interior again changes nothing, and a smaller set has a smaller interior.
These three features define an interior operator.