In
mathematics, a
residuated Boolean algebra is a
residuated lattice whose lattice structure is that of a
Boolean algebra. Examples include Boolean algebras with the monoid taken to be conjunction, the set of all formal languages over a given alphabet S under concatenation, the set of all binary relations on a given set
X under relational composition, and more generally the power set of any equivalence relation, 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.