---
tag: BDEX7HET71
kind: concept
name: Interior algebra
msc: ["06E25", "03G05", "03B45"]
specializes: [0XADJKM699]
sources:
  - url: https://en.wikipedia.org/wiki/Interior_algebra
same_as:
  wikidata: Q10894825
  wikipedia: Interior algebra
---

## Summary

An interior algebra is a [[Boolean algebra|0XADJKM699]] with one more operation that behaves like the [[interior|0RZH6W98HK]] operation of a [[topological space|0CQHFPGJ45]].
It is the algebraic form of topology, in the same way as a Boolean algebra is the algebraic form of set operations.
Interior algebras are the algebraic models of the [[modal logic|C3RZY20ETK]] S4, where the interior operation interprets "necessarily".
The dual notion, with a closure operation instead, is called a closure algebra.

## Explanations

### L1 human

Take a topological space, such as the real line.
For any set of points you can take its interior, the points that have some room around them inside the set.
The interior of the interval $[0, 1]$ is $(0, 1)$.
Taking the interior twice gives nothing new, and the interior of an intersection is the intersection of the interiors.
An interior algebra keeps only these rules and forgets the points.