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Every associative algebra is obviously alternative, but so too are some strictly nonassociative algebras such as the octonions.

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## Some Related Sentences

Every and associative

__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 and algebra

__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__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__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__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__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

**.**

Every and obviously

__Every__lethal viral disease presents a paradox: killing its host

**is**

__obviously__of no benefit to

**the**virus

**,**

**so**how and why did it evolve to do

**so**?

Every and alternative

__Every__year

**,**a huge gathering of young Poles meet to celebrate

**the**rock and

__alternative__music in Jarocin

**,**Żary

**,**and Kostrzyn nad Odrą and at Open ' er Festival and Off Festival

**.**

A remix album of

__Every__Man and Woman**is**a Star**,**entitled Companion**,**was issued in May 2003 and collected various early**,**__alternative__**,**unreleased or live versions from 1990 to 1993**.**__Every__year

**,**

**the**magazine gives out its Utne Independent Press Awards

**,**which honor

__alternative__and independent magazines from around

**the**world

**.**

Others offer an

__alternative__reading for**the**passage**,**for example**,**theologian C**.**H**.**Dodd suggests that it "**is**probably to be rendered "**as****,**"__Every__inspired scripture**is**also useful ..." A similar translation has been included in**the**New English Bible**,**Revised English Bible**,**and**as**a footnoted__alternative__in**the**New Revised Standard Version**.**__Every__Tuesday pooja

**is**done and on

__alternative__Tuesday Devi Pathri Darshana

**is**performed where

**the**woes of devotees

**are**heard by Devidarshana Pathri and solutions

**are**given which come true

**.**

* Sting-" Message in a Bottle ", " Driven To Tears ", "

__Every__Breath You Take " ( with__alternative__lyrics ) ( HP 21: 45 )1.188 seconds.