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Topic: Injective

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Injective function

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	Injective cogenerator - Wikipedia, the free encyclopedia
		In category theory, the concept of an injective cogenerator is drawn from examples such as Pontryagin duality.
		For example, the integers are a generator of the category of abelian groups (since every abelian group is a quotient of a free abelian group).
		The Tietze extension theorem can be used to show that an interval is an injective cogenerator in a category of topological spaces subject to separation axioms.
	en.wikipedia.org /wiki/Injective_cogenerator (551 words)

	Bijection, injection and surjection - Wikipedia, the free encyclopedia
		A function is injective (one-to-one) if each image is mapped to by at most one element of the domain.
		A note on terminology: a one-to-one function is injective, but may fail to be surjective, while a one-to-one correspondence is both injective and surjective.
		A function f : A → B is injective if and only if A is empty or f is left-invertible, that is, there is a function g: B → A such that g o f = identity function on A.
	www.wikipedia.org /wiki/Injective_function (1039 words)

	Opposite Algebras (Site not responding. Last check: 2007-11-06)
		Furthermore the dual of a projective OA-module is an injective A-module and the dual of a projective OA-resolution of a module M is an A-injective resolution of the dual of M. Subsections
		Injective hulls, and injective resolutions of a module are computed by taking the projective cover or projective resolution of the dual module over the opposite algebra and then again taking the dual to retrieve modules or complexes over the original algebra.
		The complex giving the minimal injective resolution of the module M together with the inclusion homomorphism from M into its injective hull.
	magma.maths.usyd.edu.au /magma/htmlhelp/text858.htm (747 words)

	Injection (mathematics) : Injective
		This function is injective, since given arbitrary real numbers
		A function is bijective if and only if it is both injective and surjective.
		In other words, injective functions are precisely the monomorphisms in the category of sets.
	www.wordlookup.net /in/injective.html (418 words)

	Overtoun M. Jenda
		Gorenstein injective dimension and Tor-depth of modules (with Edgar Enochs), Archiv Mathematick, 72 (1999), 107-117.
		Gorenstein injective and flat dimensions (with Edgar Enochs), Mathematica Japonica 44 (1996), 261-268.
		Injective resolvents and preenvelopes, Quaestiones Mathematicae 9 (1986), 301-309.
	www.math.buffalo.edu /mad/PEEPS/jenda_overtoun_m.html (759 words)

	Injection - Wikipédia
		Ces desiderata n'ont rien d'incompatible, et l'application sera dite bijective si elle est à la fois injective et surjective, et qu'en d'autres termes chaque touriste a sa chambre et chaque chambre son touriste.
		Une fonction est bijective si et seulement si elle est à la fois injective et surjective.
		En d'autres termes, les fonctions injectives sont précisément les monomorphismes de la catégorie des ensembles.
	fr.wikipedia.org /wiki/Injection (302 words)

	Math 417 Homework 1
		Thus f is injective and surjective, hence bijective.
		Thus fg is surjective and injective, hence bijective.
		Thus the set of injective functions from A to A is the same as the set of bijective functions.
	www.math.unl.edu /~bharbour/M417Spr03/M417Hmwk1Sols.html (1306 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		Many categories have injective objects, but their properties depend on what families of subobjects are allowed.
		Injective spaces are also easily proved to be closed under arbitrary products and continuous retracts, which facts provide many other examples.
		Perhaps it is not so obvious, however, that injective spaces are also closed under the formation of function spaces, once the space of continuous functions is given the right topology; indeed the category of injective spaces and continuous functions is a cartesian closed category.
	www.stanford.edu /~sommer/Scott.html (287 words)

	[No title]
		On relatively injective comodules, * is defined by (E(n)E(n) E(n)N) = BPBP BP* N. Since every comodule is a kernel of a map between relatively injective comodule* s, and is left exact, this determines * in general.
		Proposition 2.11 implies that M0 = p-1BP* is the injective hull of BP* as a BP*BP -comodule.
		0, and colimIt is injective by Proposition 2.8.
	hopf.math.purdue.edu /Hovey-Strickland/derived-ln.txt (6734 words)

	Injective and surjective (Site not responding. Last check: 2007-11-06)
		The complementary property to injectiveness is surjectiveness: a surjective function is one such that every object of the value sort is the value of the function for some argument or arguments.
		An injective function, then, is one-one while a surjective function maps the argument sort(s) onto the value sort.
		Some interesting dyadic functions are not injective in the above sense, but are injective (or surjective, or bijective) in the left argument place, the right argument place or both.
	www.cs.jcu.edu.au /~marianne/Finder/node15.html (273 words)

	Lexicon as an injective map (Site not responding. Last check: 2007-11-06)
		In a lexical map that is injective any word has exactly one meaning; every word has a meaning and two different words have different meanings.
		However the additional restriction of one meaning per word is connected only to the question of existing synonyms, and for the reason given above it is assumed here that only partial synonymy is possible and consequently the restriction is reasonable and does not change the principle character of semasiological lexica.
		An injective lexical map is therefore a good formal description of a semasiological lexicon.
	coral.lili.uni-bielefeld.de /~ttrippel/datr/node19.html (273 words)

	Formal Foundations of Computer Science 1 -- 6.1 Further Notions (Site not responding. Last check: 2007-11-06)
		For an injective function, the image of the function domain is at least as large as the domain; for a surjective function, the domain is at least as large as the image.
		If a function is injective and surjective, the domain and its image thus have the same size.
		is neither injective (because the horizontal lines with positive vertical coordinates are intersected twice) nor surjective (because the horizontal lines with negative coordinates are not intersected at all).
	www.risc.uni-linz.ac.at /education/courses/formal/report/index_44.html (476 words)

	Composition; Injective and Surjective Functions (Site not responding. Last check: 2007-11-06)
		A function f: A -> B is said to be injective (also known as one-to-one) if no two elements of A map to the same element in B. Note that some elements of B may remain unmapped in an injective function.
		Another way to describe an injective function is to say that no element of the codomain is hit more than once by the mapping.
		An important observation about injective functions is this: An injection from A to B means that the cardinality of A must be no greater than the cardinality of B
	www.mathsci.appstate.edu /~dap/classes/1100/sect3_3.html (679 words)

	AMCA: On Semigroups of Partial Injective Endomorphisms of Graphs by Inna Terekhina (Site not responding. Last check: 2007-11-06)
		By analogy with [1], the main tools of these investigations of the semigroups of all partial injective endomorphisms of graphs are canonical relations of semigroups of partial injective transformations, which are defined in these semigroups by formulas of the lower predicate language.
		This approach makes it possible to study the concrete characterization problem for the semigroups of all partial injective endomorphisms of graphs and to construct a relatively elementary interpretation of the class of graphs in the class of semigroups.
		These results can also be applied to investigation of partial injective automata the state sets and the exit sets of which endowed with algebraic structures of graphs.
	at.yorku.ca /c/a/e/c/15.htm (318 words)

	Injective spaces and the filter monad
		An injective space is a topological space with a strong extension property for continuous maps with values on it.
		In previous work we established an injectivity theorem for monads of this type, which characterizes the injective objects over a certain class of embeddings as the algebras.
		We thus obtain as a corollary that the injective spaces over subspace embeddings are the continuous lattices endowed with the Scott topology (Dana Scott, 1972).
	www.lfcs.inf.ed.ac.uk /reports/98/ECS-LFCS-98-383 (188 words)

	PlanetMath: injective module
		is an injective module if it satisfies the following equivalent conditions:
		This is version 4 of injective module, born on 2001-12-12, modified 2004-03-11.
		(Associative rings and algebras :: Modules, bimodules and ideals :: Injective modules, self-injective rings)
	planetmath.org /encyclopedia/InjectiveModule.html (81 words)

	Projective, Injective Module question... [Math] (Site not responding. Last check: 2007-11-06)
		integral domain R is an injective R-module, then R is a field.
		injective since if rx = 0 then since R is a domain, and x assumed nonzero,
		> integral domain R is an injective R-module, then R is a field.
	www.adras.com /Projective-Injective-Module-question.t448-92.html (319 words)

	Atlas: A rigid approach to injective envelopes by Martin Mathieu (Site not responding. Last check: 2007-11-06)
		Injective envelopes of C*-algebras were introduced by Hamana in the late 1970s.
		We discuss a slightly different approach which is based on choosing the correct categorical framework and on focussing on the rigidity of the injective envelope.
		This is more direct than previous approaches and allows for an easier discussion of the relations with the local and the maximal C-algebra of quotients of a C-algebra.
	atlas-conferences.com /cgi-bin/abstract/cane-78 (128 words)

	Edgar E. Enochs - Page 2
		Injective covers and resolutions, (with Overtoun Jenda), Proceedings of the joint China-Japan Ring Theory Conference, Guilin, P. China, (1993), 42 -45.
		Gorenstein injective, projective and flat dimensions over Cohen-Macaulay rings (with Overtoun Jenda), to appear in the Proceedings of the International Conference on Algebra and its Applications (Athens, Ohio 1999).
		The Gorenstein injective envelope of the residue field of a local ring (with Richard Belshoff), to appear in Comm.
	www.ms.uky.edu /~enochs/info.html (2476 words)

	Injective Cogenerator Encyclopedia Article, Definition, History, Biography (Site not responding. Last check: 2007-11-06)
		Looking For injective cogenerator - Find injective cogenerator and more at Lycos Search.
		Find injective cogenerator - Your relevant result is a click away!
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	www.karr.net /encyclopedia/Injective_cogenerator (696 words)

	categories: injective modules in a topos (Site not responding. Last check: 2007-11-06)
		A week or so ago, I asked the question about injectives in a category of modules in over a ring object in a Grothendieck topos.
		I asked whether if I is an injective module and E is an object of the topos, I^E is injective.
		So what I am asking is whether for an injective I, the induced B -o I --> A -o I is epic.
	north.ecc.edu /alsani/ct01(9-12)/msg00002.html (162 words)

	Faculty in Mathematics & CS (Site not responding. Last check: 2007-11-06)
		This theorem can also be used to show that there exists a closed, immersed, incompressible surface in the Whitehead Link Complement which remains injective if we exclude at most sixty surgeries for each of the components.
		A corollary to this result is that twist knots with a sufficiently large number of twists contain an immersed, closed, injective surface.
		Under certain conditions, we can show that the geometric sum of two (immersed) injective quasi-fuchsian surface will be a collection of injective quasi-fuchsian surfaces.
	euler.slu.edu /Dept/Faculty/bart/research.html (556 words)

	categories: Re: injective modules in a topos (Site not responding. Last check: 2007-11-06)
		This is the same as saying that the Sierpinski space is injective with respect to subspace inclusions (regular monos, if you please).
		The "internal injectivity" property in this situation is therefore that we have a retraction i U >-----------> X I U >--------> X Sigma Sigma <<-------- i Sigma but if the map I is "internal" then this is a Scott-continuous map, and we only have certain kinds of subspaces.
		The intersection problem is clearly an undesirable feature of this theory, and I believe that the "internal" injectivity is the flaw.
	north.ecc.edu /alsani/ct01(9-12)/msg00004.html (276 words)

	Función de Injective (Site not responding. Last check: 2007-11-06)
		En matemáticas, una función injective (o la función o la inyección una por) es una función que traz valores distintos de la entrada a los valores distintos de la salida.
		Por la definición, una función es bijective si y solamente si es injective y surjective.
		Es decir las funciones injective son exacto los monomorfismos en el grupo de categoría de grupos.
	www.yotor.net /wiki/es/fu/Funci%F3n%20de%20Injective.htm (265 words)

	eLibrary Project : Injective cogenerator (Site not responding. Last check: 2007-11-06)
		is very special in structure: it is pure injective module,pure-injective (also called algebraically compact), which says more or less that solving equations in ''H''
		The Tietze extension theorem can be used to show that an interval (mathematics),interval is an injective cogenerator in a category of topological spaces subject to separation axioms.
		Last Updated: Thursday 16th of June 2005 08:10:25 PM Access the sitemap.
	elibraryproject.com /info/Injective/cogenerator.html (568 words)

	[No title]
		From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik) Subject: Re: Is exponentiation injective on infinite cardinals?
		Subject: Re: Is exponentiation injective on infinite cardinals?
		It doesn't imply GCH, though; for example if for every alpha one has 2^{Aleph_alpha}=Aleph_{alpha+2}, (which is consistent with ZFC by Easton forcing), one still has that the continuum function is injective.
	www.math.niu.edu /~rusin/known-math/99/luzin_easton (1129 words)

	A mapping is sort of like a connection between two sets
		As described above, the reason why the mapping F:AàB is not an injective mapping is because F(a) = F(-a) and a ≠ -a.
		We would also have a mapping that is injective if set A were the set of all negative elements and zero.
		One final note to this page is that if a mapping is both injective and surjective, the mapping can be classified as a bijective mapping.
	students.uww.edu /muellerbt15/Mapping.htm (831 words)

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