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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.

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## 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**can

**be**obtained in this way from

**a**suitable

**topological**

**space**: see Stone's representation theorem

**for**

__Boolean__algebras

**.**

__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 is

__Every__legislator from Brasstown Bald

**to**Folkston

__is__going

**to**have his

**every**vote subjected

**to**

**the**closest scrutiny

**as**

**a**test

**of**his political allegiances

**,**not his convictions

**.**

__Every__detail in his interpretation has been beautifully thought out

**,**

**and**

**of**these I would especially cite

**the**delicious laendler touch

**the**pianist brings

**to**

**the**fifth variation ( an obvious indication that he

__is__playing with Viennese musicians )

**,**

**and**

**the**gossamer shading throughout

**.**

__Every__taxpayer

__is__well aware

**of**

**the**vast size

**of**our annual defense budget

**and**most

**of**our readers also realize that

**a**large portion

**of**these expenditures go

**for**military electronics

**.**

__Every__few days

**,**in

**the**early morning

**,**

**as**

**the**work progressed

**,**twenty men would appear

**to**push it ahead

**and**

**to**shift

**the**plank foundation that distributed its weight widely on

**the**Rotunda pavement

**,**supported

**as**it

__is__by ancient brick vaulting

**.**

__Every__dream

**,**

**and**this

__is__true

**of**

**a**mental image

**of**any type even though it may

**be**readily interpreted into its equivalent

**of**wakeful thought

**,**

__is__

**a**psychic phenomenon

**for**

**which**no explanation

__is__available

**.**

__Every__library borrower

**,**or at least those whose taste goes beyond

**the**five-cent fiction rentals

**,**knows what it

__is__

**to**hear

**the**librarian say apologetically

**,**`` I'm sorry

**,**but we don't have that book

**.**

__Every__community

**,**if it

__is__alive has

**a**spirit

**,**

**and**that spirit

__is__

**the**center

**of**its unity

**and**identity

**.**

**The**restricted principle "

__Every__partially ordered

**set**has

**a**maximal totally ordered subset "

__is__also equivalent

**to**AC over ZF

**.**

__Every__natural-born citizen

**of**

**a**foreign state who

__is__also an American citizen

**and**

**every**natural-born American citizen who

__is__

**a**citizen

**of**

**a**foreign land owes

**a**double allegiance

**,**one

**to**

**the**United States

**,**

**and**one

**to**his homeland ( in

**the**event

**of**an immigrant becoming

**a**citizen

**of**

**the**US ), or

**to**his adopted land ( in

**the**event

**of**an emigrant natural born citizen

**of**

**the**US becoming

**a**citizen

**of**another nation ).

__Every__line

**of**written text

__is__

**a**mere reflection

**of**references from any

**of**

**a**multitude

**of**traditions

**,**or

**,**

**as**Barthes puts it

**,**"

**the**text

__is__

**a**tissue

**of**quotations drawn from

**the**innumerable centres

**of**culture "; it

__is__never original

**.**

__Every__year

**,**on

**the**last Sunday in April

**,**there

__is__an ice fishing competition in

**the**frozen estuarine waters

**of**

**the**Anadyr River's mouth

**.**

__Every__

**lattice**element

**of**

**the**structure

__is__in its proper place

**,**whether it

__is__

**a**single atom or

**a**molecular grouping

**.**

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