Page "Isomorphism of categories" ¶ 13
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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, given a commutative diagram, it defines a poset category:

Conversely and Boolean
Conversely, for any topology the clopen ( i. e. closed and open ) subsets yield a Boolean algebra.
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 in rings with unity, any simple left R module arises this way.
** 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 ab 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, we have every opportunity to prevent our lives from being boring.
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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