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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.

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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__

**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

**.**

__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__

**,**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 any

__Conversely__

**,**

**a**subset R defines

**a**binary function if

**and**only if

**,**for

__any__x in X

**and**y in Y

**,**there exists

**a**unique z in Z such that

**(**x

**,**y

**,**z

**)**belongs

**to**R

**.**

__Conversely__

**the**equilibrium position

**is**said

**to**

**be**" far

**to**

**the**left " if hardly

__any__product

**is**formed

**from**

**the**reactants

**.**

* The spectrum

**of**__any__bounded linear operator on**a**Banach space**is****a**nonempty compact subset**of****the**complex**numbers**C**.**__Conversely__**,**__any__compact subset**of**C arises in this manner**,**as**the**spectrum**of**some bounded linear operator**.**__Conversely__

**,**

__any__surjection between sets determines

**a**partition on its domain

**,**

**the**

**set**

**of**preimages

**of**singletons in

**the**codomain

**.**

__Conversely__

**,**Derek Bickerton

**and**others argue that

**the**advent

**of**abstract words provided

**a**mental basis for analyzing higher-order relations

**,**

**and**that

__any__communication system that remotely resembles human language utterly relies on cognitive architecture that co-evolved alongside language

**.**

__Conversely__

**,**

**a**person may feel like he belongs

**to**one state without having

__any__legal relationship

**to**it

**.**

__Conversely__

**,**

**the**interior

**of**

**the**box has

**a**constant

**,**

**zero**potential energy .< ref

**>**Actually

**,**

__any__constant

**,**finite potential

**can**

**be**specified within

**the**box

**.**

__Conversely__

**,**extremely thin jewelry

**can**cause

**the**same tearing in what

**is**commonly referred

**to**as

**the**" cheese cutter effect ", either during sudden torsion or

**over**

**a**long period

**of**wearing

**,**especially if

**the**thin jewelry bears

__any__weight

**.**

__Conversely__

**,**

__any__Hermitian positive semidefinite

**matrix**M

**can**

**be**written as M =

**A**

**<**sup >*</ sup

**>**

**A**; this

**is**

**the**Cholesky decomposition

**.**

__Conversely__

**,**if

**a**double sharp were added

**to**

__any__other note not containing

**a**sharp or flat as indicated

**by**

**the**key signature then

**the**note will

**be**raised two semitones or

**a**whole step with respect

**to**

**the**chromatic scale

**.**

__Conversely__

**,**supplementation with L-phenylalanine

**,**L-tyrosine

**,**L-DOPA

**,**or

__any__

**of**

**the**cofactors will increase their respective concentrations

**.**

__Conversely__

**,**

__any__choice

**of**μ

**,**full rank

**matrix**U

**,**

**and**positive diagonal entries Λ

**<**

**sub**

**>**i

**</**

**sub**

**>**yields

**a**non-singular multivariate normal distribution

**.**

It

**is****a**fact that**the****ring****is****a**principal ideal**ring**; that**is****,**for__any__ideal I in**,**there exists an integer**n**in I such that every element**of**I**is****a**multiple**of****n****.**__Conversely__**,****the****set****of**all multiples**of**an arbitrary integer**n****is**necessarily an ideal**,****and****is**usually denoted**by****(****n**).

Conversely and ring

This

__ring__**is****the**endomorphism__ring__**of****A****.**__Conversely__**,**every__ring__**(**with identity**)****is****the**endomorphism__ring__**of**some object in some preadditive**category****.**__Conversely__

**,**

**given**

**a**Boolean

__ring__R

**,**

**we**define

**the**join operation

**by**

**a**' b =

**a**+ b + ab

**,**

**and**

**the**meet operation as

**multiplication**

**.**

__Conversely__

**,**

**the**valuation on

**a**discrete valuation

__ring__

**can**

**be**extended

**to**

**a**valuation on

**the**quotient field giving

**a**discrete valued field

**,**whose associated discrete valuation

__ring__

**is**just

**.**

__Conversely__

**,**if this identity holds in

**a**

__ring__R for all pairs

**of**elements

**a**

**and**b

**of**

**the**

__ring__

**,**then R

**is**commutative

**.**

__Conversely__

**,**Haystacks was an accomplished wrestler

**,**

**and**he consequently took

**the**majority

**of**

**the**decisions

**,**many

**by**countout

**(**as Humphrey often could not get himself back into

**the**

__ring__

**by**

**the**count

**of**20 ).

Statement 4 then implies that

**a**basis**of**M / mM lifts**to****a**minimal**set****of**generators**of**M**.**__Conversely__**,**every minimal**set****of**generators**of**M**is**obtained in this way**,****and****any**two such sets**of**generators are related**by**an invertible**matrix**with entries in**the**__ring__**.**
If f

**is**unramified at**a**point x**,**then f**is**quasi-finite at x**.**__Conversely__**,**if f**is**quasi-finite at x**,****and**if also**,****the**local__ring__**of**x in**the**fiber f**<**sup >− 1**</**sup >( f**(**x )),**is****a**field**and****a**finite separable extension**of**κ**(**f**(**x )), then f**is**unramified at x .< ref**>**EGA IV**<****sub****>**4**</****sub**>, Théorème 17**.**4**.**1 .</ ref**>**__Conversely__

**,**every distributive lattice

**is**isomorphic

**to**

**a**

__ring__

**of**sets ; in

**the**case

**of**finite distributive lattices

**,**this

**is**Birkhoff's representation theorem

**and**

**the**sets may

**be**taken as

**the**lower sets

**of**

**a**partially ordered

**set**

**.**

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