Page "Commutative diagram" ¶ 16
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## Some Related Sentences

Conversely and given
Conversely, a current of one ampere is one coulomb of charge going past a given point per second:
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, suppose we are given ( U, V, W ) satisfying
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
Conversely, the mapping cylinder is the homotopy pushout of the diagram where and.
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.
where e is the identity in G. Conversely, given a section s one defines a trivialization Φ by
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, a crossed module with surjective boundary defines a central extension.
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, in a preadditive category, every binary equaliser can be constructed as a kernel.
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.
Some specific kinds of commutative rings are given with the following chain of class inclusions:
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.
First, given a commutative diagram of the form
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 >.
The problem is to determine if a black box group, given by k generators, is commutative.
The dilation is commutative, also given by:.
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.

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