Boolean algebra (structure)

For the elementary syntax and axiomatics of the subject, see Boolean algebra (logic).

In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets.

Boolean lattice of subsets

1 Definition
2 Examples
3 Homomorphisms and isomorphisms
4 Boolean rings, ideals and filters
5 Representing Boolean algebras
6 Axiomatics for Boolean algebras
7 Generalizations
8 History
9 See also
10 References
11 External links

Definition

A Boolean algebra is a six-tuple consisting of a set A, equipped with two binary operations $\and$ (called "meet" or "and"), $\lor$ (called "join" or "or"), a unary operation $\lnot$ (called "complement" or "not") and two elements 0 and 1 (sometimes denoted by ⊥ and $\top$ ), such that for all elements a, b and c of A, the following axioms hold:

a ∨ (b ∨ c) = (a ∨ b) ∨ c	a ∧ (b ∧ c) = (a ∧ b) ∧ c	associativity
a ∨ b = b ∨ a	a ∧ b = b ∧ a	commutativity
a ∨ (a ∧ b) = a	a ∧ (a ∨ b) = a	absorption
a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c)	a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c)	distributivity
a ∨ ¬a = 1	a ∧ ¬a = 0	complements

A Boolean algebra with only one element is called a trivial Boolean algebra or a degenerate Boolean algebra. (Some authors require 0 and 1 to be distinct elements in order to exclude this case.)

It follows from the first three pairs of axioms above (associativity, commutativity and absorption) that

a = a ∧ b if and only if a ∨ b = b.

The relation ≤ defined by a ≤ b if and only if the above equivalent conditions hold, is a partial order with least element 0 and greatest element 1. The meet a ∧ b or the join a ∨ b of two elements coincides with their infimum or supremum, respectively, with respect to ≤.

As in every lattice, the relations ∧ and ∨ satisfy the first three pairs of axioms above; the fourth pair is just distributivity. Since the complements in a distributive lattice are unique, to define the involution ¬ it suffices to define ¬a as the complement of a.

The set of axioms is self-dual in the sense that if one exchanges ∨ with ∧ and 0 with 1 in an axiom, the result is again an axiom. Therefore by applying this operation to a Boolean algebra (or Boolean lattice), one obtains another Boolean algebra with the same elements; it is called its dual.

Examples

The simplest non-trivial 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

a	0	1
¬a	1	0

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)

Starting with the propositional calculus with κ sentence symbols, form the Lindenbaum algebra (that is, the set of sentences in the propositional calculus modulo tautology). This construction yields a Boolean algebra. It is in fact the free Boolean algebra on κ generators. A truth assignment in propositional calculus is then a Boolean algebra homomorphism from this algebra to {0,1}.

Given any linearly ordered set with a least element, the interval algebra is the smallest algebra of subsets of L containing all of the half-open intervals [a, b) such that a is in L and b in either in L or equal to ∞. Interval algebras are useful in the study of Lindenbaum-Tarski algebras; every countable BA is isomorphic to an interval algebra.

The power set (set of all subsets) of any given nonempty 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 two Boolean algebras A and B is a function f : A → B such that for all a, b in A:

f(a $\lor$ b) = f(a) $\lor$ f(b)

f(a $\land$ b) = f(a) $\land$ 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 full subcategory of the category of lattices.

A homomorphism is called a monomorphism, epimorphism or isomorphism if it is injective, surjective or bijective, respectively. The inverse map of an isomorphism is also an isomorphism.

Boolean rings, ideals and filters

Every Boolean algebra (A, $\land$ , $\lor$ ) gives rise to a ring (A, +, *) by defining a + b = (a $\land$ ¬b) $\lor$ (b $\land$ ¬a) (this operation is called symmetric difference in the case of sets and XOR in the case of logic) and a * b = a $\land$ 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 $\lor$ y = x + y - xy and x $\land$ 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 $\lor$ y in I and for all a in A we have a $\land$ 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 $\land$ 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 $\land$ y in p and for all a in A if a $\lor$ 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 clopen sets in some (compact totally disconnected Hausdorff) topological space.

Axiomatics for Boolean algebras

In 1933, the American mathematician Edward Vermilye Huntington (1874–1952) set out the following elegant axiomatization for Boolean algebra. It requires just one binary operation + and a unary functional symbol n, to be read as 'complement', which satisfy the following laws:

Commutativity: x + y = y + x.
Associativity: (x + y) + z = x + (y + z).
Huntington equation: n(n(x) + y) + n(n(x) + n(y)) = x.

Herbert Robbins immediately asked: If the Huntington equation is replaced with its dual, to wit:

4. Robbins Equation: n(n(x + y) + n(x + n(y))) = x,

do (1), (2), and (4) form a basis for Boolean algebra? Calling (1), (2), and (4) a Robbins algebra, the question then becomes: Is every Robbins algebra a Boolean algebra? This question (which came to be known as the Robbins conjecture) remained open for decades, and became a favorite question of Alfred Tarski and his students.

In 1996, William McCune at Argonne National Laboratory, building on earlier work by Larry Wos, Steve Winker, and Bob Veroff, answered Robbins's question in the affirmative: Every Robbins algebra is a Boolean algebra. Crucial to McCune's proof was the automated reasoning program EQP he designed. For a simplification of McCune's proof, see Dahn (1998).

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 $a\land x=0$ and $a\lor x=b$ . Defining $a\setminus b$ as the unique x such that $(a\land b)\lor x=a$ and $(a\land b)\land x=0$ , we say that the structure $(B,\land,\lor,\setminus,0)$ is a generalized Boolean algebra, while $(B,\lor,0)$ 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.

History

The term "Boolean algebra" honors George Boole (1815–1864), a self-educated English mathematician. He introduced the algebraic system of logic initially in a small pamphlet, The Mathematical Analysis of Logic, published in 1847 in response to an ongoing public controversy between Augustus De Morgan and William Hamilton, and later as a more substantial book, The Laws of Thought, published in 1854. Boole's formulation differs from that described above in some important respects. For example, conjunction and disjunction in Boole were not a dual pair of operations. Boolean algebra emerged in the 1860s, in papers written by William Jevons and Charles Peirce. To the 1890 Vorlesungen of Ernst Schröder we owe the first systematic presentation of Boolean algebra and distributive lattices. The first extensive treatment of Boolean algebra in English is A. N. Whitehead's 1898 Universal Algebra. Boolean algebra as an axiomatic algebraic structure in the modern axiomatic sense begins with a 1904 paper by Edward Vermilye Huntington. Boolean algebra came of age as serious mathematics with the work of Marshall Stone in the 1930s, and with Garrett Birkhoff's 1940 Lattice Theory. In the 1960s, Paul Cohen, Dana Scott, and others found deep new results in mathematical logic and axiomatic set theory using offshoots of Boolean algebra, namely forcing and Boolean-valued models.

References

Brown, Stephen; Vranesic, Zvonko (2002), Fundamentals of Digital Logic with VHDL Design (2nd ed.), McGraw–Hill, ISBN 978-0-07-249938-4 . See Section 2.5.

Cori, Rene; Lascar, Daniel (2000), Mathematical Logic: A Course with Exercises, Oxford University Press, ISBN 978-0-19-850048-3 . See Chapter 2.

Dahn, B. I. (1998), "Robbins Algebras are Boolean: A Revision of McCune's Computer-Generated Solution of the Robbins Problem", Journal of Algebra 208: 526–532, ISSN 0021-8693 .

Halmos, Paul (1963), Lectures on Boolean Algebras, Van Nostrand, ISBN 978-0-387-90094-0 .

Halmos, Paul; Givant, Steven (1998), Logic as Algebra, Dolciani Mathematical Expositions, no. 21, Mathematical Association of America, ISBN 978-0-88385-327-6 .

Huntington, E. V. (1933), "New sets of independent postulates for the algebra of logic", Transactions of the American Mathematical Society 35: 274–304, ISSN 0002-9947 .

Huntington, E. V. (1933), "Boolean algebra: A correction", Transactions of the American Mathematical Society 35: 557–558, ISSN 0002-9947 .

Mendelson, Elliott (1970), Boolean Algebra and Switching Circuits, Schaum's Outline Series in Mathematics, McGraw–Hill, ISBN 978-0-07-041460-0 .

Monk, J. Donald; Bonnet, R., eds. (1989), Handbook of Boolean Algebras, North-Holland, ISBN 978-0-444-87291-3 . In 3 volumes. (Vol.1:ISBN 978-0-444-70261-6, Vol.2:ISBN 978-0-444-87152-7, Vol.3:ISBN 978-0-444-87153-4)

Stoll, R. R. (1963), Set Theory and Logic, W. H. Freeman, ISBN 978-0-486-63829-4 . Reprinted by Dover Publications, 1979.

External links

Boolean Algebra from AllAboutCircuits
Stanford Encyclopedia of Philosophy: "The Mathematics of Boolean Algebra," by J. Donald Monk.
McCune W., 1997. Robbins Algebras Are Boolean JAR 19(3), 263--276
"Boolean Algebra" by Eric W. Weisstein, Wolfram Demonstrations Project, 2007.

A monograph available free online:

Burris, Stanley N.; Sankappanavar, H. P., 1981. A Course in Universal Algebra. Springer-Verlag. ISBN 3-540-90578-2.
Eric W. Weisstein, Boolean Algebra at MathWorld.