---
tag: GQK4DSA11K
kind: concept
name: Relation algebra
msc: ["03G15"]
specializes: [0XADJKM699]
sources:
  - url: https://en.wikipedia.org/wiki/Relation_algebra
same_as:
  wikidata: Q5965969
  wikipedia: Relation algebra
---

## Summary

A relation algebra is a [[Boolean algebra|0XADJKM699]] with two more operations and a constant.
They behave like [[composition|0PYSR20HGF]] and [[converse|054K3HNN5Z]] of [[binary relations|05AVARPXA2]] and like the identity relation.
Alfred Tarski gave the axioms in the 1940s, building on the calculus of relations of De Morgan, Peirce and Schröder.
The main example is the algebra of all binary relations on a set.
Relation algebras give an algebraic form of a fragment of first-order logic.
Tarski and Givant showed that set theory can be formalized in this setting.

## Explanations

### L1 human

Relations such as "is a parent of" can be combined in several ways.
Composing "parent of" with itself gives "grandparent of", and taking its converse gives "child of".
They also form a Boolean algebra: one can take "parent or sibling", "not a parent", and so on.
A relation algebra keeps the laws that these operations obey and studies them abstractly, without a set of people in the background.