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Conversely, given a groupoid G in the algebraic sense, let G < sub > 0 </ sub > be the set of all elements of the form x * x < sup >− 1 </ sup > with x varying through G and define G ( x * x < sup >-1 </ sup >, y * y < sup >-1 </ sup >) as the set of all elements f such that y * y < sup >-1 </ sup > * f * x * x < sup >-1 </ sup > exists.

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## Some Related Sentences

Conversely and given

__Conversely__

**,**if

**a**Boolean ring A is

__given__

**,**we can turn it into

**a**Boolean algebra by defining

**x**∨

**y**:=

**x**+

**y**+

**(**

**x**·

**y**)

**and**

**x**∧

**y**:=

**x**·

**y**

**.**

__Conversely__

**,**there are other stars

**that**never rise above

**the**horizon

**,**

**as**seen from any

__given__point on

**the**Earth's surface

**(**except exactly on

**the**equator ).

__Conversely__

**,**

__given__central idempotents

**a**

**<**

**sub**

**>**

**1**

**</**

**sub**>,...,

**a**

**<**

**sub**

**>**n

**</**

**sub**

**>**

**in**R

**that**are pairwise orthogonal

**and**have sum

**1**

**,**then R is

**the**direct sum

**of**

**the**rings Ra

**<**

**sub**

**>**

**1**

**</**

**sub**>,…, Ra

**<**

**sub**

**>**n

**</**

**sub**>.

Given

**a**field ordering ≤**as****in**Def**1****,****the****elements****such****that****x**≥**0**forms**a**positive cone**of**F**.**__Conversely__**,**__given__**a**positive cone P**of**F**as****in**Def 2**,**one can associate**a**total ordering ≤<**sub****>**P**</****sub****>**by setting**x**≤**y**to mean**y**−**x**∈ P**.**This total ordering ≤<**sub****>**P**</****sub****>**satisfies**the**properties**of**Def**1****.**__Conversely__

**,**only

**the**wiki users interested

**in**

**a**

__given__project need look at its associated wiki pages

**,**

**in**contrast to high-traffic mailing lists which may burden subscribers

**with**many messages

**,**regardless

**of**their relevance

**.**

__Conversely__

**,**

__given__

**a**harmonic function

**,**it is

**the**real part

**of**an analytic function

**,**

**(**at least locally ).

Each convex

**set**containing X must**(**by**the**assumption**that**it is convex ) contain**all**convex combinations**of**points**in**X**,**so**the****set****of****all**convex combinations is contained**in****the**intersection**of****all**convex sets containing X**.**__Conversely__**,****the****set****of****all**convex combinations is itself**a**convex**set**containing X**,**so it also contains**the**intersection**of****all**convex sets containing X**,****and**therefore**the**sets__given__by these two definitions must**be**equal**.**__Conversely__

**,**surveys conducted among living donors postoperatively

**and**

**in**

**a**period

**of**five years following

**the**procedure have shown extreme regret

**in**

**a**majority

**of**

**the**donors

**,**who said

**that**

__given__

**the**chance to repeat

**the**procedure

**,**they would not

**.**

__Conversely__

**,**

__given__an ordered tree

**,**

**and**conventionally draw

**the**root at

**the**top

**,**then

**the**child nodes

**in**an ordered tree can

**be**drawn left-to-right

**,**yielding an essentially unique planar embedding

**(**up to embedded homotopy

**,**i

**.**e., moving

**the**edges

**and**nodes without crossing ).

__Conversely__

**,**

**the**influence

**of**

**the**data at any

__given__point on

**the**initial line propagates

**with**

**the**finite velocity c: there is no effect outside

**a**triangle

**through**

**that**point whose sides are characteristic curves

**.**

__Conversely__

**,**

__given__any harmonic function

**in**two dimensions

**,**it is

**the**real part

**of**an analytic function

**,**at least locally

**.**

__Conversely__

**,**if

**(**A

**,**m

**,**e

**,**inv ) is

**a**group object

**in**one

**of**those categories

**,**then m necessarily coincides

**with**

**the**

__given__operation on A

**,**e is

**the**inclusion

**of**

**the**

__given__identity element on A

**,**inv is

**the**inversion operation

**and**A

**with**

**the**

__given__operation is an abelian group

**.**

__Conversely__

**,**if

**the**Turing Machine is expected polynomial-time

**(**for any

__given__

**x**), then

**a**considerable fraction

**of**

**the**runs must

**be**polynomial-time bounded

**,**

**and**

**the**coin sequence used

**in**

**such**

**a**run will

**be**

**a**witness

**.**

__Conversely__

**,**

__given__any ring

**,**we can

**form**

**a**category by taking objects A

**<**

**sub**

**>**n

**</**

**sub**

**>**indexed by

**the**

**set**

**of**natural numbers

**(**including zero )

**and**letting

**the**hom-set

**of**morphisms from to

**be**

**the**

**set**of-by-matrices over

**,**

**and**where composition is

__given__by matrix multiplication

**.**

__Conversely__

**,**it is hard to believe

**,**

__given__

**the**length

**and**intensity

**of**

**the**struggle between Máel Sechnaill

**and**Brian

**,**

**that**

**the**High King would surrender his title without

**a**fight

**.**

__Conversely__

**,**an algorithm to test for solvability

**in**arbitrary integers could

**be**used to test

**a**

__given__equation for solvability

**in**natural numbers by applying

**that**supposed algorithm to

**the**equation obtained from

**the**

__given__equation by replacing each unknown by

**the**sum

**of**

**the**squares

**of**four new unknowns

**.**

__Conversely__

**,**

**a**disease

**that**is easily transmitted but has

**a**short duration might spread widely during 2002 but is likely to have

**a**low prevalence at any

__given__point

**in**2003

**(**due to its short duration ) but

**a**high incidence during 2002

**(**

**as**many people develop

**the**disease ).

Conversely and G

__Conversely__

**,**if F is left adjoint to

__G__

**,**

**and**

__G__is naturally isomorphic to

__G__′ then F is also left adjoint to

__G__′.

As stated earlier

**,**an adjunction between categories C**and**D gives rise to**a**family**of**universal morphisms**,**one for each object**in**C**and**one for each object**in**D**.**__Conversely__**,**if there**exists****a**universal morphism to**a**functor__G__: C → D from every object**of**D**,**then__G__has**a**left adjoint**.**__Conversely__

**,**every Polish space is homeomorphic to

**a**

__G__

**<**

**sub**

**>**δ

**</**

**sub**>- subset

**of**

**the**Hilbert cube

**.**

__Conversely__

**,**one can show

**that**

**given**

**a**manifold M

**and**

**a**principal H-bundle P over M

**,**

**and**

**a**1-form η

**with**these properties

**,**then P is locally isomorphic

**as**an H-bundle to

**the**principal homogeneous bundle

__G__→

__G__/ H

**.**

The principal connection α on Q can

**be**recovered from**the****form**η by taking Q to**be****the**associated bundle P ×<**sub****>**H**</****sub****>**__G__**.**__Conversely__**,****the****form**η can**be**recovered from α by pulling back along**the**inclusion P ⊂ Q**.**
Given then

**a**normal extension L**of**K**,****with**automorphism group Aut**(**L / K ) =__G__**,****and**containing α**,**any element g**(**α ) for g**in**__G__will**be****a**conjugate**of**α**,**since**the**automorphism g sends roots**of**p to roots**of**p**.**__Conversely__any conjugate β**of**α is**of**this**form**:**in**other words**,**__G__acts transitively on**the**conjugates**.**
A section Γ

**of**dπ: TP /__G__→ TM**such****that**Γ: TM → TP /__G__is**a**linear morphism**of**vector bundles over M**,**can**be**identified**with****a**principal connection**in**P**.**__Conversely__**,****a**principal connection**as**defined above gives rise to**such****a**section Γ**of**TP /__G__**.**__Conversely__

**,**

**given**

**a**left kG module M

**,**then M is

**a**k vector space

**,**

**and**multiplication

**with**an element g

**of**

__G__yields

**a**k-linear automorphism

**of**M

**(**since g is invertible

**in**kG ), which describes

**a**group homomorphism

__G__→ GL

**(**M ).

Conversely and algebraic

)

__Conversely__**,**it is often helpful to use**the**geometry**of**Cartesian coordinates**in**two or three dimensions to visualize__algebraic__relationships between two or three**of**many non-spatial variables**.**__Conversely__

**,**if is

**a**separable

__algebraic__extension

**,**

**and**if L is any intermediate field

**,**then

**and**are separable extensions

**.**

Conversely and sense

__Conversely__

**,**British English favours fitted

**as**

**the**past tense

**of**fit generally

**,**whereas

**the**preference

**of**American English is more complex: AmEng prefers fitted for

**the**metaphorical

__sense__

**of**having made an object " fit "

**(**i

**.**e., suited ) for

**a**purpose ;

**in**spatial transitive contexts

**,**AmEng uses fitted for

**the**

__sense__

**of**having made an object conform to an unchanged object

**that**it surrounds

**(**e

**.**g., " fitted X around Y ") but fit for

**the**

__sense__

**of**having made an object conform to an unchanged object

**that**surrounds it

**(**e

**.**g., " fit X into Y ");

**and**for

**the**spatial senses

**(**both intransitive

**and**transitive )

**of**having been matching

**with**respect to contour

**,**

**with**no alteration

**of**either object implied

**,**AmEng prefers fit (" The clothes fit

**.**

__Conversely__

**,**most realists

**(**specifically

**,**indirect realists ) hold

**that**perceptions or

__sense__data are caused by mind-independent objects

**.**

__Conversely__

**,**

**the**rejection

**of**syncretism

**,**usually

**in**

**the**name

**of**" piety "

**and**" orthodoxy ", may help to generate

**,**bolster or authenticate

**a**

__sense__

**of**uncompromised cultural unity

**in**

**a**well-defined minority or majority

**.**

__Conversely__

**,**

**in**

**the**category

**of**rings

**,**there are no kernels

**in**

**the**category-theoretic

__sense__; indeed

**,**this category does not even have zero morphisms

**.**

"

__Conversely__**,****in****the**positive__sense__**,**it would refer to " having it both ways " or " having**the**best**of**both worlds**.**__Conversely__

**,**any operator satisfying

**the**above properties defines

**a**connection on E

**and**

**a**connection

**in**this

__sense__is also known

**as**

**a**covariant derivative on E

**.**

__Conversely__

**,**those who had

**a**difficult time during middle adulthood may look back

**and**feel

**a**

__sense__

**of**despair

**.**

__Conversely__

**,**an Ehresmann connection H ⊂ TP

**(**or v: TP → V ) on P defines

**a**principal G-connection ω if

**and**only if it is G-equivariant

**in**

**the**

__sense__

**that**

**.**

__Conversely__

**,**

**the**topological charge only makes

__sense__

**in**

**the**ordered phase

**and**not at

**all**

**in**

**the**disordered phase

**,**because

**in**some hand-waving way there is

**a**" topological condensate "

**in**

**the**disordered phase which randomizes

**the**field from point to point

**.**

__Conversely__

**,**synthetic languages often combine many lexical morphemes into single words

**,**making it difficult to boil them down to

**the**traditional

__sense__

**of**words found more easily

**in**analytic languages ; this is especially difficult for polysynthetic languages

**,**

**such**

**as**Inuktitut

**and**Ubykh

**,**where entire sentences may consist

**of**

**a**single word

**.**

for

**all**compactly supported functions ƒ**and**h**.**__Conversely__**,**() characterizes Δ completely**,****in****the**__sense__**that**it is**the**only operator**with**this property**.**__Conversely__

**and**to some surprisingly

**,**

**the**word may

**be**used to mean " approve

**of**," especially

**in**an official

__sense__

**.**

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