booleanization

Booleanization is the process of associating to a given bounded distributive lattice L a Boolean algebra B that contains a copy of L and can be viewed as the Boolean closure of L. In essence, it adjoins complements to the lattice elements to form a structure akin to classical propositional logic. The construction is used in lattice theory, topology, and logic, and there are several equivalent ways to realize it.

In the standard duality approach for bounded distributive lattices (Stone and Priestley dualities), L corresponds to a Priestley space X. The Booleanization B is canonically isomorphic to the Boolean algebra of clopen up-sets of X (equivalently, the algebra of regular open sets of X). The natural embedding i: L → B sends each a in L to the up-set {x in X : x ≤ a}. This embedding has the property that any lattice embedding of L into a Boolean algebra C factors uniquely through i into a homomorphism B → C, making B the Boolean reflection of L.

Variants and scope: In different contexts, Booleanization is defined for other algebraic structures, such as rings with idempotents or Heyting algebras, yielding a smallest Boolean object that contains the original structure. The resulting Boolean algebra often encodes the “Boolean part” or classical content of the original lattice.

Examples and use: The Booleanization of the lattice of open sets O(X) of a topological space X is the Boolean algebra of regular open sets RO(X). In logic, Booleanization relates to moving from intuitionistic or many-valued frameworks toward a classical Boolean semantics on the underlying lattice of propositions.

See also: Stone representation, Priestley duality, regular open sets, Boolean algebra, lattice theory.