A residuated Boolean algebra is an algebraic structure (L, ∧, ∨, ¬, 0, 1, •, I, \, /) such that An equivalent signature better suited to the relation algebra application is (L, ∧, ∨, ¬, 0, 1, •, I, ▷, ◁) where the unary operations x\ and x▷ are intertranslatable in the manner of De Morgan's laws via and dually /y and ◁y as with the residuation axioms in the residuated lattice article reorganized accordingly (replacing z by ¬z) to read This De Morgan dual reformulation is motivated and discussed in more detail in the section below on conjugacy.
Among residuated lattices, Boolean algebras are special by virtue of having a complementation operation ¬.
This permits an alternative expression of the three inequalities in the axiomatization of the two residuals in terms of disjointness, via the equivalence x ≤ y ⇔ x∧¬y = 0.
Abbreviating x∧y = 0 to x # y as the expression of their disjointness, and substituting ¬z for z in the axioms, they become with a little Boolean manipulation Now ¬(x\¬z) is reminiscent of De Morgan duality, suggesting that x\ be thought of as a unary operation f, defined by f(y) = x\y, that has a De Morgan dual ¬f(¬y), analogous to ∀xφ(x) = ¬∃x¬φ(x).
The cancellation step in the above is not possible for Example 3, which therefore is not a relation algebra, x˘ being uniquely determined as x▷I.