> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/400-abstract-algebra/terms/relation-algebra.md). # Relation algebra In mathematics and abstract algebra, a **relation algebra** is a *residuated Boolean algebra* expanded with an involution called *converse* (unary operation). The motivating example of a relation algebra is the algebra `2↑X²` of all binary relations on a set `X`, that is, subsets of the cartesian square `X²`, with `R • S` interpreted as the usual composition of binary relations `R` and `S`, and with the converse of `R` as the converse relation. * Relation algebra emerged in the 19th-century work of Augustus De Morgan and Charles Peirce, which culminated in the algebraic logic of Ernst Schröder. * The equational form of relation algebra was developed by Alfred Tarski and his students, starting in the 1940s. * Tarski and Givant (1987) applied relation algebra to a variable-free treatment of axiomatic set theory, with the implication that mathematics founded on set theory could itself be conducted without variables. A relation algebra (𝙇, ∧, ∨, ⁻, 0, 1, •, ˘, 𝑰) is an *algebraic structure* equipped with the Boolean operations of * conjunction `x ∧ y` * disjunction `x ∨ y` * negation `x⁻` * Boolean constant `0` * Boolean constant `1` * relational operation of composition `x • y` * relational operation of converse `x˘` * relational constant `I` such that these operations and constants satisfy certain equations constituting an axiomatization of a *calculus of relations*. Roughly, a relation algebra is to a *system of binary relations* on a set containing the empty (0), complete (1), and identity (I) relations and closed under these 5 operations as a group is to a system of permutations of a set containing the identity permutation and closed under composition and inverse. However, the first order theory of relation algebras is not complete for such systems of binary relations.