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Conversely, given a commutative diagram, it defines a poset category:

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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 commutative

__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 and diagram

In two dimensions

**,**the finite reflection groups are the dihedral groups**,**which are generated by reflection in two lines that form an angle of and correspond to the Coxeter__diagram____Conversely__**,**the cyclic point groups in two dimensions are not generated by reflections**,**and indeed contain no reflections – they are however subgroups of index 2 of**a**dihedral group.

Conversely and defines

__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__

**,**Psychology

__defines__bottom-up processing as an approach wherein there is

**a**progression from the individual elements to the whole.

__Conversely__

**,**

**a**system of n quantities v < sup > i </ sup > that transform like the coordinates x < sup > i </ sup > on V

__defines__

**a**contravariant vector.

Thus

**,**b < sub > q </ sub > is**a**symmetric bilinear form over K with matrix A.__Conversely__**,**any symmetric bilinear form b__defines__**a**quadratic form__Conversely__the building can be recovered from the BN pair

**,**so that every BN pair canonically

__defines__

**a**building.

__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__

**,**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__

**,**any such one form

__defines__( via pullback )

**a**G-equivariant horizontal 1-form on P

**,**and the space of principal G-connections is an affine space for this space of 1-forms.

__Conversely__

**,**singly even dimensional manifolds have

**a**skew-symmetric nondegenerate bilinear form on their middle dimension ; if one

__defines__

**a**quadratic refinement of this to

**a**quadratic form ( as on

**a**framed manifold ), one obtains the Arf invariant as

**a**mod 2 invariant.

Conversely and category

__Conversely__

**,**

**a**particular map between particular objects may be called an unnatural isomorphism ( or " this isomorphism is not natural ") if the map cannot be extended to

**a**natural transformation on the entire

__category__.

__Conversely__

**,**many theorems that hold in universal algebra do not generalise all the way to

__category__theory.

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__

**,**in the

__category__of rings

**,**there are no kernels in the category-theoretic sense ; indeed

**,**this

__category__does not even have zero morphisms.

__Conversely__

**,**the creation by an author of an imaginary country — such as Ruritania or Graustark — does not automatically transform that imaginary country into

**a**fantasy world

**,**even if the location would be impossible in reality owing to

**a**lack of land to contain

**it**; but such Ruritanian romances may be pushed toward the

__category__of fantasy worlds by the introduction of

**,**say

**,**witches and wise women

**,**where

**it**is not clear if their magic is effectual.

given and commutative

This extension of the definition is also compatible with the generalization for

__commutative__rings__given__below.
* The ring of formal power series over

**a**__commutative__ring R can be thought of as the inverse limit of the rings**,**indexed by the natural numbers as usually ordered**,**with the morphisms from to__given__by the natural projection.
If R is

**a**__given____commutative__ring**,**then the set of all polynomials in the variable X whose coefficients are in R forms the polynomial ring**,**denoted R. The same holds true for several variables.
In an abelian

**category**( such as the**category**of abelian groups or the**category**of vector spaces over**a**__given__field ), consider**a**__commutative__**diagram****:**
Consider the following

__commutative__**diagram**in any abelian**category**( such as the**category**of abelian groups or the**category**of vector spaces over**a**__given__field ) or in the**category**of groups.
Polynomial interpolation is finding

**a**polynomial whose values ( not coefficients ) agree with**a**__given__sequence ; the Hilbert polynomial is an abstract case of this in__commutative__algebra.
In other words

**,**assume that p = p ( x < sub > 1 </ sub >,..., x < sub > n </ sub >) is**a**non-zero polynomial in n variables**,**and that there is**a**__given__monomial order on the set of all (" monic ") monomials in these variables**,**i. e.,**a**total order of the free__commutative__monoid generated by x < sub > 1 </ sub >,..., x < sub > n </ sub >, with the unit as lowest element**,**and respecting multiplication.
If the

__given__ring is__commutative__**,****a**group ring is also referred to as**a**group algebra**,**for**it**is indeed an algebra over the__given__ring.
More generally

**,**this likewise applies to the square matrices whose entries are elements of any other__given__semiring S**,**and the semiring is generally non-commutative even though S may be__commutative__.
Given an n-dimensional formal group law F over R and

**a**__commutative__R-algebra S**,**we can form**a**group F ( S ) whose underlying set is N < sup > n </ sup > where N is the set of nilpotent elements of S. The product is__given__by using F to multiply elements of N < sup > n </ sup >; the point is that all the formal power series now converge because they are being applied to nilpotent elements**,**so there are only**a**finite number of nonzero terms.
Any affine group scheme is the spectrum of

**a**__commutative__Hopf algebra ( over**a**base S**,**this is__given__by the relative spectrum of an O < sub > S </ sub >- algebra ).
For example

**,**there is**a**duality between__commutative__rings and affine schemes**:**to every__commutative__ring A there is an affine spectrum**,**Spec A**,**conversely**,**__given__an affine scheme S**,**one gets back**a**ring by taking global sections of the structure sheaf O < sub > S </ sub >.
In computational algebraic geometry and computational

__commutative__algebra**,**Buchberger's algorithm is**a**method of transforming**a**__given__set of generators for**a**polynomial ideal into**a**Gröbner basis with respect to some monomial order.
A frame

**,**with its multiplication__given__by the meet operation**,**is**a**typical example of**a**strictly two-sided__commutative__quantale.
Any

__commutative__algebra is**a**supercommutative algebra if__given__the trivial gradation ( i. e. all elements are even ).
* Circulant matrices form

**a**__commutative__algebra**,**since for any two__given__circulant matrices and**,**the sum is circulant**,**the product is circulant**,**and.
Specifically

**,**__given__**a**sequence of cohomology groups H < sup > k </ sup >( X ; R ) on X with coefficients in**a**__commutative__ring R ( typically R is Z < sub > n </ sub >, Z**,**Q**,**R**,**or C ) one can define the cup product**,**which takes the form
The proof just

__given__indicates the scope of the identity in abstract algebra**:****it**will hold in any__commutative__ring R.1.150 seconds.