---
tag: KCG48CWY69
kind: concept
name: Residuated Boolean algebra
sources:
  - wikipedia: en:12435391@r1352956591
    locator: Residuated Boolean algebra
    note: adopted
    mention: wikipedia://en/12435391
    sections: [Summary]
same_as:
  wikidata: Q7315522
  wikipedia: Residuated Boolean algebra
inventory: [W4FV457732]
---

## Summary

In mathematics, a **residuated Boolean algebra** is a residuated lattice whose lattice structure is that of a [[Boolean algebra|0XADJKM699]]. Examples include Boolean algebras with the [[monoid|0N6R902FG1]] taken to be conjunction, the set of all formal languages over a given alphabet $\Sigma$ under concatenation, the set of all [[binary relations|05AVARPXA2]] on a given set $X$ under relational composition, and more generally the [[power set|00E0YZCERH]] of any [[equivalence relation|0G8VEZ3WTK]], again under relational composition. The original application was to relation algebras as a finitely axiomatized generalization of the binary relation example, but there exist interesting examples of residuated Boolean algebras that are not relation algebras, such as the language example.