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Then an element e of S is called a left identity if e * a = a for all a in S, and a right identity if a * e = a for all a in S. If e is both a left identity and a right identity, then it is called a two-sided identity, or simply an identity.

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## Some Related Sentences

Then and element

__Then__

**is**

**a**group whose

**identity**

__element__

**is**The group inverse

**of**

**an**arbitrary group

__element__

**is**the function inverse

*****Let the index set I

**of**

**an**inverse system ( X < sub > i </ sub >, f < sub > ij </ sub >) have

**a**greatest

__element__m

**.**

__Then__the natural projection π < sub > m </ sub >: X → X < sub > m </ sub >

**is**

**an**isomorphism

**.**

__Then__

**,**after the pump energy stored

**in**the laser medium has approached the maximum possible level

**,**the introduced loss mechanism ( often

**an**electro-or acousto-optical

__element__)

**is**rapidly removed (

**or**that occurs by itself

**in**

**a**passive device ), allowing lasing to begin which rapidly obtains the stored energy

**in**the gain medium

**.**

For f

**a**real polynomial**in**x**,****and****for**any**a****in**such**an**algebra define f (**a**) to be the__element__**of**the algebra resulting from the obvious substitution**of****a**into f**.**__Then__**for**any two such polynomials f**and**g**,**we have that ( fg ) (**a**)__Then__

**,**

**in**1846

**,**the

__element__ilmenium was claimed to have been discovered

**,**but later was determined to be impure niobium

**.**

__Then__

**in**1908

**,**the Japanese chemist Masataka Ogawa found evidence

**in**the mineral thorianite

**,**which he thought indicated the presence

**of**

__element__43

**.**

Suppose

**a**partially ordered set P has the property that every chain ( i**.****e****.**totally ordered subset ) has**an**upper bound**in**P**.**__Then__the set P contains at least one maximal__element__**.**
Suppose

**a**non-empty partially ordered set P has the property that every non-empty chain has**an**upper bound**in**P**.**__Then__the set P contains at least one maximal__element__**.**__Then__p + q

**=**( p < sub > 1 </ sub >+ q < sub > 1 </ sub >, p < sub > 2 </ sub >+ q < sub > 2 </ sub >); since p < sub > 1 </ sub >

**=**p < sub > 2 </ sub >

**and**q < sub > 1 </ sub >

**=**q < sub > 2 </ sub >,

**then**p < sub > 1 </ sub > + q < sub > 1 </ sub >

**=**p < sub > 2 </ sub > + q < sub > 2 </ sub >, so p + q

**is**

**an**

__element__

**of**W

**.**

# Let p

**=**( p < sub > 1 </ sub >, p < sub > 2 </ sub >) be**an**__element__**of**W**,**that**is****,****a**point**in**the plane such that p < sub > 1 </ sub >**=**p < sub > 2 </ sub >,**and**let c be**a**scalar**in**R**.**__Then__cp**=**( cp < sub > 1 </ sub >, cp < sub > 2 </ sub >); since p < sub > 1 </ sub >**=**p < sub > 2 </ sub >,**then**cp < sub > 1 </ sub >**=**cp < sub > 2 </ sub >, so cp**is****an**__element__**of**W**.**__Then__

**,**

**for**example

**,**(−∞, 1 )

**and**( 0

**,**∞) would be

**in**the topology generated by

**S**

**,**being unions

**of**

**a**single base

__element__

**,**

**and**so their intersection ( 0

**,**1 ) would be as well

**.**

__Then__

**if**we intersect each

__element__

**of**B with Y

**,**the resulting collection

**of**sets

**is**

**a**base

**for**the subspace Y

**.**

"

__Then__with her**right**hand she would sprinkle water at the head**of**the child**and**say**,**" Behold this__element__without whose assistance no mortal being can survive**.**__Then__the union ( written as +)

**and**the concatenation ( written as ·)

**of**two elements

**of**A again belong to A

**,**

**and**so does the Kleene star operation applied to any

__element__

**of**A

**.**

Let B be

**a**complex Banach algebra containing**a**unit**e****.**__Then__we define the spectrum σ ( x ) (**or**more explicitly σ < sub > B </ sub >( x ))**of****an**__element__x**of**B to be the set**of**those complex numbers λ**for**which λe − x**is**not invertible**in**B**.**__Then__the film was rewound

**,**

**and**everything except the foreground

__element__matted out so that the foreground

__element__would now photograph

**in**the previously blacked out area

**.**

__Then__the

__element__c

**=**ba

**is**nilpotent (

**if**non-zero ) as c < sup > 2 </ sup >

**=**( ba )< sup > 2 </ sup >

**=**b ( ab )

**a**

**=**0

**.**

Then and e

__Then__

**,**removing the catalyst would also result

**in**reaction

**,**producing energy ; i

**.**

__e__

**.**the addition

**and**its reverse process

**,**removal

**,**would

**both**produce energy

**.**

__Then__X

**is**compact

**if**

**and**only

**if**X

**is**

**a**complete lattice ( i

**.**

__e__

**.**

**all**subsets have suprema

**and**infima ).

__Then__

**is**

**a**derivation

**and**

**is**linear

**,**i

**.**

__e__.,

**and**

**,**

**and**

**a**Lie algebra homomorphism

**,**i

**.**

__e__.,

**,**but

**it**

**is**not always

**an**algebra homomorphism

**,**i

**.**

__e__

**.**the

**identity**does not hold

**in**general

**.**

__Then__

**,**once this claim ( expressed

**in**the previous sentence )

**is**proved

**,**

**it**will suffice to prove " φ

**is**either refutable

**or**satisfiable " only

**for**φ's belonging to the class C

**.**Note also that

**if**φ

**is**provably equivalent to ψ ( i

**.**

__e__., ( φ ≡ ψ )

**is**provable ),

**then**

**it**

**is**indeed the case that " ψ

**is**either refutable

**or**satisfiable " → " φ

**is**either refutable

**or**satisfiable " ( the soundness theorem

**is**needed to show this ).

__Then__

**,**~ 150

**,**000 years later ( i

**.**

__e__

**.**around 50

**,**000 years ago ), sub-groups

**of**this population began to expand our species ' range to regions outside

**of**

**,**

**and**( later ) within

**,**this continent ( Tishkoff

**,**1996 ).

__Then__

**a**general definition

**of**isomorphism that covers the previous

**and**many other cases

**is**:

**an**isomorphism

**is**

**a**morphism that has

**an**inverse

**,**i

**.**

__e__

**.**there exists

**a**morphism with

**and**

**.**

__Then__the data are passed through high-performance restartable dithering engine which

**is**used regardless

**of**monitor bit depth

**,**i

**.**

__e__

**.**also

**for**24 bits per pixel colour

**.**

Consider some set P

**and****a**binary relation ≤ on P**.**__Then__≤**is****a**preorder**,****or**quasiorder**,****if****it****is**reflexive**and**transitive**,**i**.**__e__.,**for****all****a****,**b**and**c**in**P**,**we have that:__Then__

**,**

**in**1905

**,**to explain the photoelectric effect ( 1839 ), i

**.**

__e__., that shining light on certain materials can function to eject electrons from the material

**,**Albert Einstein postulated

**,**based on Planck ’ s quantum hypothesis

**,**that light itself consists

**of**individual quantum particles

**,**which later came to be

**called**photons ( 1926 ).

__Then__

**it**was pointed out that " seeing the most submarines " depended not only on the number

**of**submarines present

**,**but also on the number

**of**eyes looking ; i

**.**

__e__., patrol density

**.**

__Then__any vector

**in**R < sup > 3 </ sup >

**is**

**a**linear combination

**of**

__e__< sub > 1 </ sub >,

__e__< sub > 2 </ sub >

**and**

__e__< sub > 3 </ sub >.

__Then__these dummies could be selected with mouse

**,**

**and**next character from the password "

__e__"

**is**typed

**,**which replaces the dummies " asdfsd ".

__Then__

**it**

**is**used as: General Strike

**of**

**a**city

**,**i

**.**

__e__., " General Strike

**in**Florence ",

**or**

**a**General Strike

**in**

**a**whole country

**or**province

**,**

**for**the purpose

**of**gaining political rights

**,**i

**.**

__e__., the

**right**to vote ; as

**in**Belgium

**,**

**or**Sweden

**.**

__Then__

**,**the markedness

**of**the ruler comes into play:

**it**

**is**" anchored " at point A

**,**

**and**slided

**and**rotated until one mark

**is**at point C

**,**

**and**one at point D

**,**i

**.**

__e__., CD

**=**AB

**.**

Then and S

__Then__came

**an**even more shocking confession: according to the CIA document

**,**al-Faruq said two senior al-Qaeda officials

**,**Abu Zubaydah

**and**Ibn al-Shaykh al-Libi

**,**had ordered him to ' plan large-scale attacks against U

**.**

__S__

**.**interests

**in**Indonesia

**,**Malaysia

**,**( the ) Philippines

**,**Singapore

**,**Thailand

**,**Taiwan

**,**Vietnam

**and**Cambodia

**.**

__Then__

**,**when the U

**.**

__S__

**.**Army Air Forces on the Marianas Islands ran out

**of**conventional thermite incendiary bombs

**for**its B-29 Superfortresses to drop on Japanese cities

**,**its top commanders

**,**such as General Curtis E

**.**LeMay turned to napalm bombs to continue its fire raids on the large Japanese cities

**.**

__Then__N < sub > x </ sub >

**is**

**a**directed set

**,**where the direction

**is**given by reverse inclusion

**,**so that

__S__≥ T

**if**

**and**only

**if**

__S__

**is**contained

**in**T

**.**For

__S__

**in**N < sub > x </ sub >, let x < sub >

__S__</ sub > be

**a**point

**in**

__S__

**.**

__Then__( x < sub >

__S__</ sub >)

**is**

**a**net

**.**

__Then__

**,**

**it**

**is**undecidable to determine whether the language decided by

**an**arbitrary Turing machine lies

**in**

__S__

**.**

In 1986

**,****a**screening**of**the entire Monkees television series by MTV led to renewed interest**in**the group**,**followed by**a**single (" That Was__Then__**,**This Is Now " reached number 20 on the Billboard Hot 100**in**the U**.**__S__.),**a**20th Anniversary Tour**,****a**greatest hits album**and****a**brand new LP**,**Pool It!__Then__

**,**

**in**1966

**,**the U

**.**

__S__

**.**Congress created

**a**National Council

**for**Marine Resources

**and**Engineering Development

**.**

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