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  "body": "\n## Summary\n\nIn mathematics, a **residuated Boolean algebra** is a residuated lattice whose lattice structure is that of a [[Boolean algebra|0XADJKM699]]. Examples include Boolean algebras with the [[monoid|0N6R902FG1]] taken to be conjunction, the set of all formal languages over a given alphabet $\\Sigma$ under concatenation, the set of all [[binary relations|05AVARPXA2]] on a given set $X$ under relational composition, and more generally the [[power set|00E0YZCERH]] of any [[equivalence relation|0G8VEZ3WTK]], 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.\n",
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