Relation Algebra

A relation algebra (L, ∧, ∨, −, 0, 1, •, I, ) is an algebraic structure equipped with the Boolean operations of conjunction x∧y, disjunction x∨y, and negation x−, the Boolean constants 0 and 1, the relational operations of composition x•y and converse x, and the relational constant I, such that these operations and constants satisfy certain equations constituting an axiomatization of relation algebras. 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 five operations as a group is to a system of permutations of a set containing the identity permutation and closed under composition and inverse.

Following Jónsson and Tsinakis (1993) it is convenient to define additional operations x◁y = x•y, and, dually, x▷y = x•y . Jónsson and Tsinakis showed that I◁x = x▷I, and that both were equal to x. Hence a relation algebra can equally well be defined as an algebraic structure (L, ∧, ∨, −, 0, 1, •, I, ◁, ▷). The advantage of this signature over the usual one that a relation algebra can then be defined in full simply as a residuated Boolean algebra for which I◁x is an involution, that is, I◁(I◁x) = x . The latter condition can be thought of as the relational counterpart of the equation 1/(1/x) = x for ordinary arithmetic reciprocal, and some authors use reciprocal as a synonym for converse.

Since residuated Boolean algebras are axiomatized with finitely many identities, so are relation algebras. Hence the latter form a variety, the variety RA of relation algebras. Expanding the above definition as equations yields the following finite axiomatization.