Boolean algebra (structure)

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, or its elements can be viewed as generalized truth values. It is also a special case of a De Morgan algebra and a Kleene algebra (with involution).
Every Boolean algebra gives rise to a Boolean ring, and vice versa, with ring multiplication corresponding to conjunction or meet ∧, and ring addition to exclusive disjunction or symmetric difference. However, the theory of Boolean rings has an inherent asymmetry between the two operators, while the axioms and theorems of Boolean algebra express the symmetry of the theory described by the duality principle.
[Image:Hasse diagram of powerset of 3.svg|thumb|right|250px|Boolean lattice of subsets]

History

The term "Boolean algebra" honors George Boole, a self-educated English mathematician. He introduced the algebraic system 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, 9th Baronet|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 Sanders Peirce.
The first systematic presentation of Boolean algebra and distributive lattices is owed to the 1890 Vorlesungen of Ernst Schröder. 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 V. 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 set theory">Set (mathematics)">set theory using offshoots of Boolean algebra, namely forcing and Boolean-valued models.

Definition

A Boolean algebra is a set, equipped with two binary operations , , a unary operation and two elements and in , such that for all elements, and of, the following axioms hold:
Note, however, that the absorption law and even the associativity law can be excluded from the set of axioms as they can be derived from the other axioms.
A Boolean algebra with only one element is called a trivial Boolean algebra or a degenerate Boolean algebra.
It follows from the last three pairs of axioms above, or from the absorption axiom, that
The relation defined by if these equivalent conditions hold, is a partial order with least element 0 and greatest element 1. The meet and the join of two elements coincide with their infimum and supremum, respectively, with respect to ≤.
The first four pairs of axioms constitute a definition of a bounded lattice.
It follows from the first five pairs of axioms that any complement is unique.
The set of axioms is self-dual in the sense that if one exchanges with and with in an axiom, the result is again an axiom. Therefore, by applying this operation to a Boolean algebra, one obtains another Boolean algebra with the same elements; it is called its dual.

Examples

The simplest non-trivial Boolean algebra, the two-element Boolean algebra, has only two elements, and, and is defined by the rules:

The power set of any given nonempty set forms a Boolean algebra, an algebra of sets, with the two operations and . The smallest element 0 is the empty set and the largest element is the set itself.

The set of all subsets of that are either finite or cofinite is a Boolean algebra and an algebra of sets called the finite–cofinite algebra. If is infinite then the set of all cofinite subsets of, which is called the Fréchet filter, is a free ultrafilter on. However, the Fréchet filter is not an ultrafilter on the power set of.
Starting with the propositional calculus with sentence symbols, form the Lindenbaum algebra. This construction yields a Boolean algebra. It is in fact the free Boolean algebra on generators. A truth assignment in propositional calculus is then a Boolean algebra homomorphism from this algebra to the two-element Boolean algebra.
Given any linearly ordered set with a least element, the interval algebra is the smallest Boolean algebra of subsets of containing all of the half-open intervals such that is in and is either in or equal to. Interval algebras are useful in the study of Lindenbaum–Tarski algebras; every countable Boolean algebra is isomorphic to an interval algebra.
For any natural number, the set of all positive divisors of, defining if divides, forms a distributive lattice. This lattice is a Boolean algebra if and only if is square-free. The bottom and the top elements of this Boolean algebra are the natural numbers and, respectively. The complement of is given by. The meet and the join of and are given by the greatest common divisor and the least common multiple of and, respectively. The ring addition is given by. The picture shows an example for. As a counter-example, considering the non-square-free, the greatest common divisor of 30 and its complement 2 would be 2, while it should be the bottom element 1.
Other examples of Boolean algebras arise from topological spaces: if is a topological space, then the collection of all subsets of that are both open and closed forms a Boolean algebra with the operations and .
If is an arbitrary ring then its set of central idempotents, which is the set

becomes a Boolean algebra when its operations are defined by and.

Homomorphisms and isomorphisms

A homomorphism between two Boolean algebras and is a function such that for all, in :
It then follows that for all in. The class of all Boolean algebras, together with this notion of morphism, forms a full subcategory of the category of lattices.
An isomorphism between two Boolean algebras and is a homomorphism with an inverse homomorphism, that is, a homomorphism such that the composition is the identity function on, and the composition is the identity function on. A homomorphism of Boolean algebras is an isomorphism if and only if it is bijective.

Boolean rings

Every Boolean algebra [|gives rise] to a ring by defining and. The zero element of this ring coincides with the 0 of the Boolean algebra; the multiplicative identity element of the ring is the of the Boolean algebra. This ring has the property that for all in ; rings with this property are called Boolean rings.
Conversely, if a Boolean ring is given, we can turn it into a Boolean algebra by defining and. Since these two constructions are inverses of each other, we can say that every Boolean ring arises from a Boolean algebra, and vice versa. Furthermore, a map 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; in fact the categories are isomorphic.
Hsiang gave a rule-based algorithm to check whether two arbitrary expressions denote the same value in every Boolean ring.
More generally, Boudet, Jouannaud, and Schmidt-Schauß gave an algorithm to solve equations between arbitrary Boolean-ring expressions. Employing the similarity of Boolean rings and Boolean algebras, both algorithms have applications in automated theorem proving.

Ideals and filters

An ideal of the Boolean algebra is a nonempty subset such that for all, in we have in and for all in we have in. This notion of ideal coincides with the notion of ring ideal in the Boolean ring. An ideal of is called prime if and if in always implies in or in. Furthermore, for every we have that, and then if is prime we have or for every. An ideal of is called maximal if and if the only ideal properly containing is itself. For an ideal, if and, then or is contained in another proper ideal. Hence, such an is not maximal, and therefore the notions of prime ideal and maximal ideal are equivalent in Boolean algebras. Moreover, these notions coincide with ring theoretic ones of prime ideal and maximal ideal in the Boolean ring.
The dual of an ideal is a filter. A filter of the Boolean algebra is a nonempty subset such that for all, in we have in and for all in we have in. The dual of a maximal ''ideal in a Boolean algebra is ultrafilter. Ultrafilters can alternatively be described as 2-valued morphisms from to the two-element Boolean algebra. The statement every filter in a Boolean algebra can be extended to an ultrafilter is called the ultrafilter lemma and cannot be proven in Zermelo–Fraenkel set theory, if ZF is consistent. Within ZF, the ultrafilter lemma is strictly weaker than the axiom of choice. The ultrafilter lemma has many equivalent formulations: every Boolean algebra has an ultrafilter, every ideal in a Boolean algebra can be extended to a prime ideal'', etc.

Representations

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 is isomorphic to the Boolean algebra of all clopen sets in some topological space.

Axiomatics

The first axiomatization of Boolean lattices/algebras in general was given by the English philosopher and mathematician Alfred North Whitehead in 1898. It included the above axioms and additionally and. In 1904, the American mathematician Edward V. Huntington gave probably the most parsimonious axiomatization based on,,, even proving the associativity laws. He also proved that these axioms are independent of each other.
In 1933, Huntington set out the following elegant axiomatization for Boolean algebra. It requires just one binary operation and a unary functional symbol, to be read as 'complement', which satisfy the following laws:
Herbert Robbins immediately asked: If the Huntington equation is replaced with its dual, to wit:
do,, and form a basis for Boolean algebra? Calling,, and a Robbins algebra, the question then becomes: Is every Robbins algebra a Boolean algebra? This question 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 computer program EQP he designed. For a simplification of McCune's proof, see Dahn.
Further work has been done for reducing the number of axioms; see Minimal axioms for Boolean algebra.

Generalizations

Removing the requirement of existence of a unit from the axioms of Boolean algebra yields "generalized Boolean algebras". Formally, a distributive lattice is a generalized Boolean lattice, if it has a smallest element and for any elements and in such that, there exists an element such that and. Defining as the unique such that and, we say that the structure is a generalized Boolean algebra, while 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 linear subspaces for separable Hilbert spaces.

Works cited

General references

. See Section 2.5
. See Chapter 2.
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. In 3 volumes.
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. Reprinted by Dover Publications, 1979.