Boolean algebra

In formal logic, mathematics and computer science, Boolean algebras, or Boolean lattices, are algebraic structures which "capture the essence" of the logical operations AND, OR and NOT as well as the corresponding set-theoretic operations intersection, union and complement.

They are named after George Boole, an English mathematician at University College Cork, who first defined them as part of a system of logic in the mid 19th century. Specifically, Boolean algebra was an attempt to use algebraic techniques to deal with expressions in the propositional calculus. Today, Boolean algebras find many applications in electronic design. They were first applied to switching by Claude Shannon in the 20th century.

The operators of Boolean algebra may be represented in various ways. Often they are simply written as AND, OR and NOT. In describing circuits, NAND (NOT AND), NOR (NOT OR) and XOR (eXclusive OR) may also be used. Mathematicians, engineers, and programmers often use + for OR and · for AND (since in some ways those operations are analogous to addition and multiplication in other algebraic structures) and represent NOT by a line drawn above the expression being negated.

Here we use another common notation with "meet" <math>\land<math> (or ^ for browsers that don't support the character) for AND, "join" <math>\lor<math> (or v) for OR, and ¬ (or ~) for NOT.

Contents

1 Definition and first consequences

2 Examples

3 Homomorphisms and isomorphisms

4 Boolean rings, ideals and filters

5 Representing Boolean algebras

6 See also

7 Links

Definition and first consequences

A Boolean algebra is a lattice (A, <math>\land<math>, <math>\lor<math>) (considered as an algebraic structure) with the following four additional properties:

bounded below: There exists an element 0, such that a <math>\lor<math> 0 = a for all a in A.
bounded above: There exists an element 1, such that a <math>\land<math> 1 = a for all a in A.
distributive law: For all a, b, c in A, (a <math>\lor<math> b) <math>\land<math> c = (a <math>\land<math> c) <math>\lor<math> (b <math>\land<math> c).
existence of complements: For every a in A there exists an element ¬a in A such that a <math>\lor<math> ¬a = 1 and a <math>\land<math> ¬a = 0.

From these axioms, one can show that the smallest element 0, the largest element 1, and the complement ¬a of any element a are uniquely determined.

Like any lattice, a Boolean algebra (A, <math>\land<math>, <math>\lor<math>) gives rise to a partially ordered set (A, ≤) by defining

a ≤ b iff a = a <math>\land<math> b

(which is also equivalent to b = a <math>\lor<math> b).

In fact one can also define a Boolean algebra to be a distributive lattice (A, ≤) (considered as a partially ordered set) with least element 0 and greatest element 1, within which every element x has a complement ¬x such that

x <math>\land<math> ¬x = 0 and x <math>\lor<math> ¬x = 1

Here <math>\land<math> and <math>\lor<math> are used to denote the infimum (meet) and supremum (join) of two elements. Again, if complements in the above sense exist, then they are uniquely determined.

The algebraic and the order theoretic perspective as usually can be used interchangeably and both are of great use to import results and concepts from both universal algebra and order theory. In many practical examples an ordering relation, conjunction, disjunction, and negation are all naturally available, so that it is straightforward to exploit this relationships.

Now one can obtain several other theorems valid in all Boolean algebras. For example, for all elements a and b of a Boolean algebra, one finds that

a <math>\land<math> 0 = 0 and a <math>\lor<math> 1 = 1,
¬1 = 0 and ¬0 = 1,
¬ ¬ a = a

and that both de Morgan's laws are valid, i.e.

¬(a <math>\lor<math> b) = (¬a) <math>\land<math> (¬b) and ¬(a <math>\land<math> b) = (¬a) <math>\lor<math> (¬b)

One can also apply general insights from duality in order theory to Boolean algebras. Especially, the order dual of every Boolean algebra, or, equivalently, the algebra obtained by exchanging <math>\land<math> and <math>\lor<math>, is also a Boolean algebra. Thus the dual version of the distributive law,

(a <math>\land<math> b) <math>\lor<math> c = (a <math>\lor<math> c) <math>\land<math> (b <math>\lor<math> c)

also holds true. In general, any law valid for Boolean algebras can be transformed into another valid dual law by exchanging 0 with 1, <math>\land<math> with <math>\lor<math>, and ≤ with ≥.

Examples

The most important Boolean algebra has only two elements, 0 and 1, and is defined by the rules:

∧	0	1
0	0	0
1	0	1

∨	0	1
0	0	1
1	1	1

It has applications in logic, interpreting 0 as false, 1 as true, ∧ as and, ∨ as or, and ¬ as not. Expressions involving variables and the Boolean operations represent statement forms, and two such expressions can be shown to be equal using the above axioms if and only if the corresponding statement forms are logically equivalent.

The two-element Boolean algebra is also used for circuit design in electrical engineering; here 0 and 1 represent the two different states of one bit in a digital circuit, typically high and low voltage. Circuits are described by expressions containing variables, and two such expressions are equal for all values of the variables if and only if the corresponding circuits have the same input-output behavior. Furthermore, every possible input-output behavior can be modeled by a suitable Boolean expression.

The two-element Boolean algebra is also important in the general theory of Boolean algebras, because an equation involving several variables is generally true in all Boolean algebras if and only if it is true in the two-element Boolean algebra (which can always be checked by a trivial brute force algorithm). This can for example be used to show that the following laws (Consensus theorems) are generally valid in all Boolean algebras:
- (a ∨ b) ∧ (¬a ∨ c) ∧ (b ∨ c) ≡ (a ∨ b) ∧ (¬a ∨ c)
- (a ∧ b) ∨ (¬a ∧ c) ∨ (b ∧ c) ≡ (a ∧ b) ∨ (¬a ∧ c)

The power set of any given set S forms a Boolean algebra with the two operations ∨ := ∪ (union) and ∧ := ∩ (intersection). The smallest element 0 is the empty set and the largest element 1 is the set S itself.

The set of all subsets of S that are either finite or cofinite is a Boolean algebra.

For any natural number n, the set of all positive divisors of n forms a distributive lattice if we write a ≤ b for a | b. This lattice is a Boolean algebra if and only if n is square-free. The smallest element 0 of this Boolean algebra is the natural number 1; the largest element 1 of this Boolean algebra is the natural number n.

Other examples of Boolean algebras arise from topological spaces: if X is a topological space, then the collection of all subsets of X which are both open and closed forms a Boolean algebra with the operations ∨ := ∪ (union) and ∧ := ∩ (intersection).

If R is an arbitrary ring and we define the set of central idempotents by
A = { e ∈ R : e² = e, ex = xe, ∀x ∈ R }
then the set A becomes a Boolean algebra with the operations e ∨ f := e + f − ef and e ∧ f := ef.

Homomorphisms and isomorphisms

A homomorphism between the Boolean algebras A and B is a function f : A → B such that for all a, b in A:

f(a <math>\lor<math> b) = f(a) <math>\lor<math> f(b)

f(a <math>\land<math> b) = f(a) <math>\land<math> f(b)

f(0) = 0

f(1) = 1

It then follows that f(¬a) = ¬f(a) for all a in A as well. The class of all Boolean algebras, together with this notion of morphism, forms a category. An isomorphism from A to B is a homomorphism from A to B which is bijective. The inverse of an isomorphism is also an isomorphism, and we call the two Boolean algebras A and B isomorphic. From the standpoint of Boolean algebra theory, they cannot be distinguished; they only differ in the notation of their elements.

Boolean rings, ideals and filters

Every Boolean algebra (A, <math>\land<math>, <math>\lor<math>) gives rise to a ring (A, +, *) by defining a + b = (a <math>\land<math> ¬b) <math>\lor<math> (b <math>\land<math> ¬a) (this operation is called "symmetric difference" in the case of sets and XOR in the case of logic) and a * b = a <math>\land<math> b. The zero element of this ring coincides with the 0 of the Boolean algebra; the multiplicative identity element of the ring is the 1 of the Boolean algebra. This ring has the property that a * a = a for all a in A; rings with this property are called Boolean rings.

Conversely, if a Boolean ring A is given, we can turn it into a Boolean algebra by defining x <math>\lor<math> y = x + y − xy and x <math>\land<math> y = xy. Since these two operations are inverses of each other, we can say that every Boolean ring arises from a Boolean algebra, and vice versa. Furthermore, a map f : A → B is a homomorphism of Boolean algebras if and only if it is a homomorphism of Boolean rings. The categories of Boolean rings and Boolean algebras are equivalent.

An ideal of the Boolean algebra A is a subset I such that for all x, y in I we have x <math>\lor<math> y in I and for all a in A we have a <math>\land<math> x in I. This notion of ideal coincides with the notion of ring ideal in the Boolean ring A. An ideal I of A is called prime if I ≠ A and if a <math>\land<math> b in I always implies a in I or b in I. An ideal I of A is called maximal if I ≠ A and if the only ideal properly containing I is A itself. These notions coincide with ring theoretic ones of prime ideal and maximal ideal in the Boolean ring A.

The dual of an ideal is a filter. A filter of the Boolean algebra A is a subset p such that for all x, y in p we have x <math>\land<math> y in p and for all a in A if a <math>\lor<math> x = a then a in p.

Representing Boolean algebras

It can be shown that every finite Boolean algebra is isomorphic to the Boolean algebra of all subsets of a finite set. Therefore, the number of elements of every finite Boolean algebra is a power of two.

Stone's celebrated representation theorem for Boolean algebras states that every Boolean algebra A is isomorphic to the Boolean algebra of all closed-open sets in some (compact totally disconnected Hausdorff) topological space.