Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Residuated Boolean algebra
open-in-new
<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a residuated Boolean algebra is a <a href="/facts/Residuated_lattice/p84UdNQ3">residuated lattice</a> whose lattice structure is that of a <a href="/facts/Boolean_algebra_(structure)/cHbo65jm">Boolean algebra</a>. Examples include Boolean algebras with the monoid taken to be conjunction, the set of all formal languages over a given alphabet Σ under concatenation, the set of all binary relations on a given set <i>X</i> under relational composition, and more generally the power set of any equivalence relation, again under relational composition. The original application was to <a href="/facts/Relation_algebra/sjRfIybo">relation algebras</a> 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. </p>