A Boolean algebra is a mathematical structure that is similar to a Boolean ring, but that is defined using the meet and join
operators instead of the usual addition and multiplication operators. Explicitly,
a Boolean algebra is the partial order on subsets
defined by inclusion (Skiena 1990, p. 207), i.e., the Boolean algebra 
of a set 
is the set of subsets of 
that can be obtained by means of a finite number of the set
operations union (OR), intersection
(AND), and complementation
(NOT) (Comtet 1974, p. 185). A Boolean algebra also
forms a lattice (Skiena 1990, p. 170), and each
of the elements of 
is called a Boolean function. There are 
Boolean
functions in a Boolean algebra of order 
(Comtet 1974, p. 186).
In 1938, Shannon proved that a two-valued Boolean algebra (whose members are most commonly denoted 0 and 1, or false and true) can describe the operation of two-valued electrical switching circuits. In modern times, Boolean algebra and Boolean functions are therefore indispensable in the design of computer chips and integrated circuits.
Boolean algebras have a recursive structure apparent in the Hasse diagrams illustrated above for Boolean algebras of orders 
, 3, 4, and 5. These figures illustrate the partition between
left and right halves of the lattice, each of which is the Boolean algebra on 
elements (Skiena 1990, pp. 169-170).
The Hasse diagram for the Boolean algebra of order 
is implemented as BooleanAlgebra[n]
in the Wolfram Language package Combinatorica`
. It is isomorphic to the 
-hypercube graph.
A Boolean algebra can be formally defined as a set 
of elements 
, 
, ... with the following properties:
1. 
has two binary operations, 
(logical AND, or "wedge")
and 
(logical OR, or "vee"),
which satisfy the idempotent laws
 |
(1)
|
the commutative laws
 |  |  |
(2)
|
 |  |  |
(3)
|
and the associative laws
 |  |  |
(4)
|
 |  |  |
(5)
|
2. The operations satisfy the absorption law
 |
(6)
|
3. The operations are mutually distributive
 |  |  |
(7)
|
 |  |  |
(8)
|
4. 
contains universal bounds 
(the empty set) and 
(the universal
set) which satisfy
 |  |  |
(9)
|
 |  |  |
(10)
|
 |  |  |
(11)
|
 |  |  |
(12)
|
5. 
has a unary operation 
of complementation, which obeys the laws
 |  |  |
(13)
|
 |  |  |
(14)
|
(Birkhoff and Mac Lane 1996).
In the slightly archaic terminology of (Bell 1986, p. 444), a Boolean algebra can be defined as a set 
of elements 
, 
, ... with binary operators 
(or 
; logical OR) and 
(or 
; logical AND) such that
1a. If 
and 
are in the set 
,
then 
is in the set 
.
1b. If 
and 
are in the set 
,
then 
is in the set 
.
2a. There is an element 
(zero) such that 
for every element 
.
2b. There is an element 
(unity) such that 
for every element 
.
3a. 
.
3b. 
.
4a. 
.
4b. 
.
5. For every element 
there is an element 
such that 
and 
.
6. There are at least two distinct elements in the set 
.
Huntington (1933ab) presented the following basis for Boolean algebra:
1. Commutativity: 
.
2. Associativity: 
.
3. Huntington axiom: 
.
H. Robbins then conjectured that the Huntington axiom could be replaced with the simpler Robbins axiom,
 |
(15)
|
The algebra defined by commutativity, associativity, and the Robbins axiom is called Robbins algebra. Computer theorem proving demonstrated that every Robbins algebra satisfies the second Winkler condition, from which it follows immediately that all Robbins algebras are Boolean (McCune, Kolata 1996).
Wells, C. F. "Boolean
Expression Manipulation."