**Generalizations**

Removing the requirement of existence of a unit from the axioms of Boolean algebra yields "generalized Boolean algebras". Formally, a distributive lattice *B* is a generalized Boolean lattice, if it has a smallest element 0 and for any elements *a* and *b* in *B* such that *a* ≤ *b*, there exists an element *x* such that a ∧ x = 0 and a ∨ x = b. Defining a ∖ b as the unique *x* such that (a ∧ b) ∨ x = a and (a ∧ b) ∧ x = 0, we say that the structure (B,∧,∨,∖,0) is a *generalized Boolean algebra*, while (B,∨,0) is a *generalized Boolean semilattice*. Generalized Boolean lattices are exactly the ideals of Boolean lattices.

A structure that satisfies all axioms for Boolean algebras except the two distributivity axioms is called an orthocomplemented lattice. Orthocomplemented lattices arise naturally in quantum logic as lattices of closed subspaces for separable Hilbert spaces.

Read more about this topic: Boolean Algebra (structure)