19: Lattices and Boolean Algebras
The axioms of a ring give structure to the operations of addition and multiplication on a set. However, we can construct algebraic structures, known as lattices and Boolean algebras, that generalize other types of operations. For example, the important operations on sets are inclusion, union, and intersection. Lattices are generalizations of order relations on algebraic spaces, such as set inclusion in set theory and inequality in the familiar number systems \({\mathbb N}\text{,}\) \({\mathbb Z}\text{,}\) \({\mathbb Q}\text{,}\) and \({\mathbb R}\text{.}\) Boolean algebras generalize the operations of intersection and union. Lattices and Boolean algebras have found applications in logic, circuit theory, and probability.