A boolean algebra B is a complete

lattice with the following properties:

For any a,b of B let

*a and b := inf{a,b}* and

*a or b:=sup{a,b}*.

- Distributivity: a and (b or c) = (a and b) or (a and c)
- Distributivity: a or (b and c) = (a or b) and (a or c)
- Existence of a negation operation
*not*: B -> B with
- ((not a) and a) or b = b
- ((not a) or a) and b = b

From these follows:

- all other calculation rules (some follow directly from the lattice)
- the unique existence of a 1 and 0 (in fact 1=sup(B) and 0=inf(B)), one could also define 1 and 0 and their properties in the definition
- boolean algebras doesn't just consist of 0 and 1, they can be rather strange in the infinite case
- there is some connection to the algebraic definition of an algebra
- in a finite boolean algebra every element can be constructed from minimal elements, the so called atoms. This is the basis for the representation of boolean functions by disjunctive normal forms

Examples: set of subsets of a set.

the set of functions from an arbitrary set into a boolean algebra