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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.

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## Some Related Sentences

Every and associative

__Every__

__associative__

**algebra**

**is**

**obviously**

**alternative**

**,**

**but**

**so**too

**are**

**some**strictly nonassociative

**algebras**such as

**the**

**octonions**

**.**

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 so

__Every__

__so__often

**the**diminishing sound of a car came under

**the**trailer as it slowed down for

**the**wreck then speeded up again as it got clear

**.**

__Every__man who dabbles in

**the**market to make a little easy money on

**the**side

**and**suffers losses could at

**the**time hardly face his wife who was wondering how her husband could be

__so__dumb

**.**

__Every__winter a kegful of this sauce was made

**and**placed at

**the**end of a row of four

**other**kegs in

**the**cellar

**,**

__so__that when its turn came

**,**it was properly mellowed

**.**

__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__Andalusian region or comarca has its own variety of this popular food

__so__there

**are**many types of gazpacho

**.**

Of Christianity he has this to say: "... it

**is**not now true for me ...__Every__believing Christian**is****,**I am sure**,**my spiritual brother ...**but**if systemically I called myself a Christian I feel that to most men I should imply too much**and**__so__tell a lie**.**__Every__Mac made between 1986

**and**1998 has a SCSI port on

**the**back

**,**making external expansion easy ; also

**,**" toaster " Compact Macs did not have easily accessible hard drive bays

**(**or

**,**in

**the**case of

**the**Mac Plus

**,**any hard drive bay at

**all**),

__so__on those models

**,**external SCSI disks were

**the**only reasonable option

**.**

__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__atom across this plane has an individual set of emission cones .</ p > < p > Drawing

**the**billions of overlapping cones

**is**impossible

**,**

__so__this

**is**a simplified diagram showing

**the**extents of

**all**

**the**emission cones combined

**.**

King Mangra was

__so__impressed that he allegedly remarked**,**"__Every__part of**the**Thai**is**blessed with venom**.**__Every__objective has a different size ring

**,**

__so__for every objective another condenser setting has to be chosen

**.**

Ebert concludes his review with

**the**following line: "__Every__time I see**the**film**,**I feel a great sadness**,**that a human imagination could be__so__limited that it sees its own extinction as a victory**.**__Every__year

**,**growers fold

**the**higher parts of

**the**plant downward

__so__

**the**plant stays at heights accessible by a standing human

**.**

The convention details

**(**amongst**other**things**)**that "__Every__driver shall keep to**the**edge of**the**carriageway appropriate to**the**direction of traffic ",**and****the**" Drivers overtaking shall do__so__on**the**side opposite to that appropriate to**the**direction of traffic ", notwithstanding**the**presence or absence of oncoming traffic**.**__Every__Duke thereafter has either died without heirs or succeeded to

**the**Crown

**,**

**and**

__so__has not been able to leave

**the**Royal Family

**.**

__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__autumn

**,**according to Plutarch

**(**Life of Lycurgus

**,**28

**,**3 – 7 ),

**the**Spartan ephors would pro forma declare war on

**the**helot population

__so__that any Spartan citizen could kill a helot without fear of blood or guilt

**(**crypteia ).

__Every__microwave oven sold has a protective interlock

__so__that it cannot be run when

**the**door

**is**open or improperly latched

**.**

__Every__ancient reference to Balaam considers him a non-Israelite

**,**a prophet

**,**

**and**

**the**son of Beor

**,**though Beor

**is**not

__so__clearly identified

**.**

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