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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.

from
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## Some Related Sentences

Every and Boolean

__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**can be obtained

**in**

**this**way from

**a**suitable topological space: see Stone's representation theorem for

__Boolean__algebras

**.**

__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 gives

__Every__LORAN chain

**in**

**the**world uses

**a**unique Group Repetition Interval

**,**

**the**number

**of**which

**,**when multiplied

**by**ten

**,**

__gives__how many microseconds pass between pulses from

**a**given station

**in**

**the**chain

**.**

__Every__year

**,**

**the**International Documentary Festival

**in**Amsterdam

**(**IDFA

**)**

__gives__an acclaimed filmmaker

**the**chance

**to**screen his or her personal Top 10 favorite films

**.**

*

__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__thing depends upon

**the**evil

**of**

**the**second order

**;**it

**is**

**this**which

__gives__

**to**such actions

**the**character

**of**crime

**,**

**and**which makes punishment necessary

**.**

__Every__adjunction 〈 F

**,**G

**,**ε

**,**η 〉

__gives__

**rise**

**to**an associated monad 〈 T

**,**η

**,**μ 〉

**in**

**the**category D

**.**

**The**functor

__Every__ticket

**is**provided

**with**

**a**bar code that

__gives__information about

**the**time

**the**visitor stepped into

**the**museum

**.**

__Every__coalgebra

**,**

**by**

**(**vector space

**)**duality

**,**

__gives__

**rise**

**to**an

**algebra**

**,**but not

**in**general

**the**other way

**.**

__Every__broadcasting company has members

**and**

**the**number

**of**members

__gives__them

**a**status that

**is**connected

**to**

**the**number

**of**hours

**of**broadcasting

**.**

**The**WBC launched

**a**website

**called**Priests Rape Boys

**in**which they criticize

**the**Roman Catholic Church because

**of**

**the**Catholic sex abuse scandal

**,**saying

**,**"

__Every__time any person

__gives__any amount

**of**money

**to**

**the**Catholic Church

**,**that person

**is**paying

**the**salary

**of**pedophile rapists

**.**

__Every__year

**,**

**the**magazine

__gives__out its Utne Independent Press Awards

**,**which honor alternative

**and**independent magazines from around

**the**world

**.**

This flat may be identified

**with****the**partition**of****the**vertices**of**into**the**connected components**of****the**subgraph formed**by**:__Every__set**of**edges having**the**same closure as__gives__**rise****to****the**same partition**of****the**vertices**,****and**may be recovered from**the**partition**of****the**vertices**,**as it consists**of****the**edges whose endpoints both belong**to****the**same set**in****the**partition**.**__Every__sequence

**of**

**A**

**,**B

**,**

**and**C without immediate repetition

**of**

**the**same one

**is**possible

**and**

__gives__an equally dense packing for spheres

**of**

**a**given radius

**.**

__Every__year

**the**American Society

**of**Architectural Illustrators

__gives__out

**the**Hugh Ferriss Memorial Prize for architectural rendering excellence

**.**

__Every__year

**,**

**the**IDFA

**,**International Documentary Festival

**in**Amsterdam

**,**

__gives__an acclaimed filmmaker

**the**chance

**to**screen his or her personal Top 10 favorite films

**.**

__Every__year

**,**she

__gives__

**a**speech

**to**cadets from throughout Civil Air Patrol at

**the**National Honor Guard Academy

**and**also

**to**

**a**gathered assembly

**of**basic cadets

**and**staff at

**the**Civil Air Patrol's Tri-Wing Encampment

**,**held

**by**Maryland Wing

**.**

__Every__episode

__gives__credits

**to**James Burrows

**,**Glen Charles

**and**Les Charles

**,**

**the**creators

**of**Cheers

**,**

**the**show where

**the**character

**of**Dr

**.**Frasier Crane originated

**.**

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