Page "Kernel (category theory)" ¶ 20
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## Some Related Sentences

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

Conversely and rings
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 >.
Conversely in rings with unity, any simple left R module arises this way.

Conversely and there
Conversely the use of true brass seems to have declined in Western Europe during this period in favour of gunmetals and other mixed alloys but by the end of the first Millennium AD brass artefacts are found in Scandinavian graves in Scotland, brass was being used in the manufacture of coins in Northumbria and there is archaeological and historical evidence for the production of brass in Germany and The Low Countries areas rich in calamine ore which would remain important centres of brass making throughout the medieval period, especially Dinant – brass objects are still collectively known as dinanterie in French.
Conversely, the Belgian Blue's meat tenderness has been argued to be just as tender because there are a large number of smaller muscle fibers.
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 existence of Bletchley Park, and of the decrypting achievements there, was never officially shared with the Soviet Union, whose war effort would have greatly benefited from regular decrypting of German messages relating to the Eastern Front.
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, corresponding to any partition of X, there exists an equivalence relation ~ on X.
Conversely, there are Grothendieck topologies which do not come from topological spaces.
That is, the motivational internalist believes that there is an internal, necessary connection between one's conviction that X ought to be done and one's motivation to do X. Conversely, the motivational externalist ( or moral externalist ) claims that there is no necessary, internal connection between moral convictions and moral motives.
Conversely, there have been a number of popes whose reign lasted less than a month.
Conversely, Baptists are underrepresented, a reflection of their quite recent expansion in numbers ; there has been only one Roman Catholic president, although they are currently the largest single denomination, and there have been no Lutheran, Pentecostal, or Latter Day Saint presidents.
Conversely, if I is not minimal, then there is a non-zero right ideal J properly contained in I. J is a right submodule of I, so I is not simple.
Conversely, if I is not maximal, then there is a right ideal J properly containing I.
Conversely, if dφ < sub > x </ sub > is an isomorphism then there is an open neighborhood U of x such that φ maps U diffeomorphically onto its image.
Conversely, the Court has approved routine warrantless seizures, for example " where there is probable cause to believe that a criminal offense has been or is being committed.
Conversely, Elijah Delmedigo ( c. 1458 – c. 1493 ), in his Bechinat ha-Dat endeavored to show that the Zohar could not be attributed to Shimon bar Yochai, arguing that if it were his work, the Zohar would have been mentioned by the Talmud, as has been the case with other works of the Talmudic period, that had bar Yochai known by divine revelation the hidden meaning of the precepts, his decisions on Jewish law from the Talmudic period would have been adopted by the Talmud, that it would not contain the names of rabbis who lived at a later period than that of Simeon ; and that if the Kabbalah was a revealed doctrine, there would have been no divergence of opinion among the Kabbalists concerning the mystic interpretation of the precepts.
Conversely, Per Christiansen found that using skull morphology allowed him to identify the subspecies krugeri, nubica, persica, and senegalensis, while there was overlap between bleyenberghi with senegalensis and krugeri.
Conversely, Psychology defines bottom-up processing as an approach wherein there is a progression from the individual elements to the whole.
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.
As stated earlier, an adjunction between categories C and D gives rise to a family of universal morphisms, one for each object in C and one for each object in D. Conversely, if there exists a universal morphism to a functor G: C → D from every object of D, then G has a left adjoint.
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, if B satisfies both of the conditions 1 and 2, then there is a unique topology on X for which B is a base ; it is called the topology generated by B.
Conversely, there are many Kohanim who do not have Cohen as a surname.
Conversely, if there is a tenfold excess of the acid with respect to the base, the ratio is 1: 10 and the pH is pKa – 1.

Conversely and are
Conversely, experimenters would consider as impressive such statements as the following, which, if they turned out to be hits, are so unusual as to be really significant:
Conversely, the vowel marks of the Tigrinya abugida and the Amharic abugida ( ironically, the original source of the term " abugida ") have been so completely assimilated into their consonants that the modifications are no longer systematic and have to be learned as a syllabary rather than as a segmental script.
Conversely, most heat-treatable alloys are precipitation hardening alloys, which produce the opposite effects that steel does.
Conversely, most realists ( specifically, indirect realists ) hold that perceptions or sense data are caused by mind-independent objects.
Conversely, many fighter aircraft, such as the F-16, are often used as ' bomb trucks ,' despite being designed for aerial combat.
Conversely, because space is expanding, and more distant objects are receding ever more quickly, light emitted by us today may never " catch up " to very distant objects.
Conversely, certain general principles ( for example, neighbourliness, or charity ) are seen to be universal and consistent.
Conversely, modern biological sciences ( including even concepts such as molecular ecology ) are intimately entwined and dependent on the methods developed through biotechnology and what is commonly thought of as the life sciences industry.
Conversely beetles are prey of various invertebrates and vertebrates including insects, fish, reptiles, birds and mammals.
Conversely, factors that favor plate columns are: presence of solids in feed, high liquid rates, large column diameters, complex columns, columns with wide feed composition variation, columns with a chemical reaction, absorption columns, columns limited by foundation weight tolerance, low liquid rate, large turn-down ratio and those processes subject to process surges.
Conversely, computer implementations are significant in applying ideas from discrete mathematics to real-world problems, such as in operations research.
Conversely, Harris maintains that the fact-value distinction is a confusion, proposing that values are really a certain kind of fact.
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, the tiny meson mass mass differences responsible for meson oscillations are often expressed in the more convenient inverse picoseconds.
Conversely, BBC critic Mark Kermode believes that " the movie industries of Britain and America are inextricably intertwined ", citing numerous examples of how Hollywood provides work to British production staff and studios, whilst Britain enables Hollywood to base their prestigious productions at UK studios.
Conversely, in functional code, the output value of a function depends only on the arguments that are input to the function, so calling a function f twice with the same value for an argument x will produce the same result f ( x ) both times.
Conversely, the bodies of people who had certain medical conditions are useful for research into those conditions.
Conversely, if the data are sparse — that is, if only K out of N Fourier coefficients are nonzero — then the complexity can be reduced to O ( K log N log ( N / K )), and this has been demonstrated to lead to practical speedups compared to an ordinary FFT for N / K > 32 in a large-N example ( N = 2 < sup > 22 </ sup >) using a probabilistic approximate algorithm ( which estimates the largest K coefficients to several decimal places ).
Conversely, rifle bullets can pierce light armor and are easier to shoot accurately, but even a carbine such as the Colt M4 is larger and / or longer than a submachine gun, making it harder to maneuver in close quarters.
Conversely, if either of these last two expressions is defined, then so is the other ( and again they are equal ).

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