[permalink] [id link]

Every Boolean algebra can be obtained in this way from a suitable topological space: see Stone's representation theorem for Boolean algebras.

from
Wikipedia

## Some Related Sentences

Every and Boolean

__Every__

__Boolean__

**algebra**( A, ∧, ∨) gives rise to

**a**ring ( A, +, ·) by defining

**a**+ b := (

**a**∧ ¬ b ) ∨ ( b ∧ ¬

**a**) = (

**a**∨ b ) ∧ ¬(

**a**∧ b ) (

**this**operation is called symmetric difference

**in**the case of sets and XOR

**in**the case of logic ) and

**a**· b :=

**a**∧ 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**

**.**

__Every__prime ideal P

**in**

**a**

__Boolean__ring R is maximal

**:**the quotient ring R / P is an integral domain and also

**a**

__Boolean__ring, so it is isomorphic to the field F < sub > 2 </ sub >, which shows the maximality of P

**.**Since maximal ideals are always prime, prime ideals and maximal ideals coincide

**in**

__Boolean__rings

**.**

__Every__

__Boolean__

**algebra**is

**a**Heyting

**algebra**when

**a**→ b is defined as usual as ¬

**a**∨ b, as is every complete distributive lattice when

**a**→ b is taken to

**be**the supremum of the set of all c

**for**which

**a**∧ c ≤ b

**.**The open sets of

**a**

**topological**

**space**form

**a**complete distributive lattice and hence

**a**Heyting

**algebra**

**.**

__Every__complemented distributive lattice has

**a**unique orthocomplementation and is

**in**fact

**a**

__Boolean__

**algebra**

**.**

__Every__finite

__Boolean__

**algebra**

**can**

**be**represented as

**a**whole power set-the power set of its set of atoms ; each element of the

__Boolean__

**algebra**corresponds to the set of atoms below it ( the join of which is the element ).

__Every__( normal )

__Boolean__

**algebra**with operators

**can**

**be**represented as

**a**field of sets on

**a**relational structure

**in**the sense that it is isomorphic to the complex

**algebra**corresponding to the field

**.**

__Every__

__Boolean__

**algebra**A has an essentially unique completion, which is

**a**complete

__Boolean__

**algebra**containing A such that every element is the supremum of some subset of A

**.**

*

__Every__subset of**a**complete__Boolean__**algebra**has**a**supremum, by definition ; it follows that every subset also has an infimum ( greatest lower bound ).

Every and algebra

__Every__associative

__algebra__is obviously alternative, but so too are some strictly nonassociative

**algebras**such as the octonions

**.**

*

__Every__real Banach__algebra__which is**a**division__algebra__is isomorphic to the reals, the complexes, or the quaternions**.**
*

__Every__unital real Banach__algebra__with no zero divisors, and**in**which every principal ideal is closed, is isomorphic to the reals, the complexes, or the quaternions**.**
*

__Every__commutative real unital Noetherian Banach__algebra__with no zero divisors is isomorphic to the real or complex numbers**.**
*

__Every__commutative real unital Noetherian Banach__algebra__( possibly having zero divisors ) is finite-dimensional**.**__Every__continuous map f

**:**X → Y induces an

__algebra__homomorphism C ( f )

**:**C ( Y ) → C ( X ) by the rule C ( f )( φ ) = φ o f

**for**every φ

**in**C ( Y ).

__Every__vector v

**in**determines

**a**linear map

**from**R to taking 1 to v, which

**can**

**be**thought of as

**a**Lie

__algebra__homomorphism

**.**

__Every__associative

__algebra__is obviously power-associative, but so are all other alternative

**algebras**( like the octonions, which are non-associative ) and even some non-alternative

**algebras**like the sedenions

**.**

__Every__random vector gives rise to

**a**probability measure on R < sup > n </ sup > with the Borel

__algebra__as the underlying sigma-algebra

**.**

__Every__finite-dimensional Hausdorff

**topological**vector

**space**is reflexive, because J is bijective by linear

__algebra__, and because there is

**a**unique Hausdorff vector

**space**topology on

**a**finite dimensional vector

**space**

**.**

__Every__Heyting

__algebra__with exactly one coatom is subdirectly irreducible, whence every Heyting

__algebra__

**can**

**be**made an SI by adjoining

**a**new top

**.**

Every and can

__Every__such subset has

**a**smallest element, so to specify our choice function we

__can__simply say that it maps each set to the least element of that set

**.**

__Every__information exchange between living organisms — i

**.**e

**.**transmission of signals that involve

**a**living sender and receiver

__can__

**be**considered

**a**form of communication ; and even primitive creatures such as corals are competent to communicate

**.**

__Every__context-sensitive grammar which does not generate the empty string

__can__

**be**transformed into an equivalent one

**in**Kuroda normal form

**.**

__Every__grammar

**in**Chomsky normal form is context-free, and conversely, every context-free grammar

__can__

**be**transformed into an equivalent one which is

**in**Chomsky normal form

**.**

__Every__module over

**a**division ring has

**a**basis ; linear maps between finite-dimensional modules over

**a**division ring

__can__

**be**described by matrices, and the Gaussian elimination algorithm remains applicable

**.**

__Every__entire function

__can__

**be**represented as

**a**power series that converges uniformly on compact sets

**.**

Group actions / representations

**:**__Every__group G__can__**be**considered as**a**category with**a**single object whose morphisms are the elements of G**.**A functor**from**G to Set is then nothing but**a**group action of G on**a**particular set, i**.**e**.****a**G-set**.**__Every__sequence

__can__, thus,

**be**read

**in**three reading frames, each of which will produce

**a**different amino acid sequence (

**in**the given example, Gly-Lys-Pro, Gly-Asn, or Glu-Thr, respectively ).

__Every__hyperbola is congruent to the origin-centered East-West opening hyperbola sharing its same eccentricity ε ( its shape, or degree of " spread "), and is also congruent to the origin-centered North-South opening hyperbola with identical eccentricity ε — that is, it

__can__

**be**rotated so that it opens

**in**the desired direction and

__can__

**be**translated ( rigidly moved

**in**the plane ) so that it is centered at the origin

**.**

__Every__holomorphic function

__can__

**be**separated into its real and imaginary parts, and each of these is

**a**solution of Laplace's equation on R < sup > 2 </ sup >.

__Every__species

__can__

**be**given

**a**unique ( and, one hopes, stable ) name, as compared with common names that are often neither unique nor consistent

**from**place to place and language to language

**.**

__Every__use of modus tollens

__can__

**be**converted to

**a**use of modus ponens and one use of transposition to the premise which is

**a**material implication

**.**

__Every__adult, healthy, sane Muslim who has the financial and physical capacity to travel to Mecca and

__can__make arrangements

**for**the care of his / her dependants during the trip, must perform the Hajj once

**in**

**a**lifetime

**.**

*

__Every__finite**topological****space**gives rise to**a**preorder on its points,**in**which x ≤ y if and only if x belongs to every neighborhood of y, and every finite preorder__can__**be**formed as the specialization preorder of**a****topological****space****in****this****way****.**
*

__Every__preorder__can__**be**given**a**topology, the Alexandrov topology ; and indeed, every preorder on**a**set is**in**one-to-one correspondence with an Alexandrov topology on that set**.**__Every__binary relation R on

**a**set S

__can__

**be**extended to

**a**preorder on S by taking the transitive closure and reflexive closure, R < sup >+=</ sup >.

0.346 seconds.