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Conversely, given a Boolean ring R, we define the join operation by a ' b = a + b + ab, and the meet operation as 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__

**,**

__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 Boolean

__Conversely__

**,**

**given**any topological space X

**,**

**the**collection of subsets of X that are clopen ( both closed

**and**open ) is

**a**

__Boolean__algebra

**.**

Conversely and ring

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 ).
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__

**,**

**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

**.**

Conversely and R

__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__

**,**if ƒ: C → C is

**a**function which is differentiable when regarded

**as**

**a**function on

__R__< sup > 2 </ sup >, then ƒ is complex differentiable if

**and**only if

**the**Cauchy – Riemann equations hold

**.**

**

__Conversely__**,**if ƒ: I →__R__is absolutely continuous**and**thus differentiable almost everywhere**,****and**satisfies | ƒ ′( x )| ≤ K for almost all x in I**,**then ƒ is Lipschitz continuous with Lipschitz constant at most K**.**__Conversely__

**,**it is possible to 2-colour

**a**K < sub > 5 </ sub > without creating any monochromatic K < sub > 3 </ sub >, showing that

__R__( 3

**,**3 ) > 5

**.**

__Conversely__

**,**if

__R__is

**a**symmetric relation over

**a**set X

**,**one can interpret it

**as**describing an undirected graph with

**the**elements of X

**as**

**the**vertices

**and**

**the**pairs in

__R__

**as**

**the**edges

**.**

__Conversely__

**,**an exhaustive firing test of

**the**M1918 Chauchat in

**.**30-06 was also carried out in 1994 near Chambersburg

**,**Pennsylvania

**,**

**by**

__R__

**.**Keller

**and**W

**.**Garofalo

**.**

__Conversely__

**,**

**the**following characterization of derived functors holds:

**given**

**a**family of functors

__R__< sup > i </ sup >: A → B

**,**satisfying

**the**above

**,**i

**.**e

**.**mapping short exact sequences to long exact sequences

**,**such that for every injective object I of A

**,**

__R__< sup > i </ sup >( I )= 0 for every positive i

**,**then these functors are

**the**right derived functors of

__R__< sup > 0 </ sup >.

Conversely and we

If neither A nor B includes

**the**idea of existence**,**then " some A are B " simply adjoins A to B**.**__Conversely__**,**if A or B do include**the**idea of existence in**the**way that " triangle " contains**the**idea " three angles equal to two right angles ", then " A exists " is automatically true**,****and**__we__have an ontological proof of A's existence**.**__Conversely__

__we__have

**a**

**b**if

**and**only if

**a**<

**b**or

**a**~

**b**

**.**This is

**the**reason for using

**the**notation ""; "≤" can be confusing for

**a**preorder that is not anti-symmetric

**,**it may suggest that

**a**≤

**b**implies that

**a**<

**b**or

**a**

__Conversely__

**,**specifying

**the**symmetry can

**define**

**the**structure

**,**or at least clarify what

__we__mean

**by**an invariant

**,**geometric language in which to discuss it ; this is one way of looking at

**the**Erlangen programme

**.**

__Conversely__

**,**

**a**scientific antirealist or instrumentalist argues that science does not aim ( or at least does not succeed ) at truth

**and**that

__we__should not regard scientific theories

**as**true

**.**

__Conversely__

**,**if

__we__pick every member of

**the**same population with property

**and**ask " what proportion of these have property?

__Conversely__

**,**

__we__understand Cleopatra at her death

**as**

**the**transcendent queen of “ immortal longings ” because

**the**container of her mortality can no longer restrain her: unlike Antony

**,**she never melts

**,**but sublimates from her very earthly flesh to ethereal fire

**and**air .”

__Conversely__

__we__come to understand Cleopatra in that

**the**container of her mortality can no longer restrain her

**.**

"

__Conversely__**,**Jeffrey Tucker of**the**Ludwig von Mises Institute has argued that " The whole scene — which anticipated so much of**the**technology__we__have today but**,**strangely**,**not email or texting — reflected**the**ethos of time:**a**love of progress**and****a**vision of**a**future that stayed on course ...__Conversely__

**,**if

__we__accept that human beings do have ( libertarian or incompatibilist ) free will

**,**then

__we__must accept that

**the**world is not entirely governed

**by**natural law

**.**

__Conversely__

**,**starting from this equivalence

__we__can recover U

**as**

**the**composite Rel → Sup → Set of

**the**forgetful functor for Sup with this embedding of Rel in Sup

**.**

__Conversely__

**,**

__we__can deduce that

**a**minority of

**the**helots were Laconian

**,**thus making this

**the**one

**and**only revolt of their history

**.**

__Conversely__

**,**

__we__will explore how insights from science can be useful

**,**not only to develop technology

**and**improve our material comfort

**,**but to reduce

**the**suffering of individuals

**,**families

**,**

**and**society

**.**

__Conversely__

**,**if

__we__say that Paul has

**the**goal of going into graduate school

**,**he may not necessarily believe that he will be able to accomplish this

**.**

__Conversely__

**,**

**given**

**a**Hopf algebra whose coalgebra structure is

**given**above

**,**

__we__can recover

**a**formal group law F from it

**.**

__Conversely__

**,**

**given**

**a**configuration of

**a**conic

**and**six lines which tangent to it in

**the**plane

**,**

__we__may

**define**

**the**double cover of

**the**plane ramified over

**the**union of these 6 lines

**.**

__Conversely__

**,**if

**the**length-scale decreases

**and**

**the**physical parameters approach fixed values

**,**then

__we__have ultraviolet fixed points

**.**

__Conversely__

**,**

**given**any element u ∈ F ( A )

__we__may

**define**

**a**natural transformation Φ: Hom ( A ,–) → F via

__Conversely__

**,**in

**the**limit

__we__have infinitely rapid

**,**or diabatic passage ;

**the**configuration of

**the**state remains unchanged:

__Conversely__

**,**if G contains

**a**proper non-trivial normal subgroup N

**,**then

**the**composition of

**the**natural surjective group homomorphism G → G / N with

**the**regular representation of G / N produces

**a**representation π of G which has kernel N

**.**Taking χ to be

**the**character of some non-trivial subrepresentation of π

**,**

__we__have

**a**character satisfying

**the**hypothesis in

**the**direct statement above

**.**

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