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# Topic: Homomorphism

 Homomorphism - Wikipedia, the free encyclopedia Homomorphism is one of the fundamental concepts in abstract algebra. In abstract algebra, a homomorphism is a map from one algebraic structure to another of the same type that preserves all the relevant structure. In that case the image of X in Y under the homomorphism f is necessarily isomorphic to X/~; this fact is one of the isomorphism theorems. en.wikipedia.org /wiki/Homomorphism   (802 words)

 Info and facts on 'Group homomorphism'   (Site not responding. Last check: 2007-11-07) For example, a homomorphism of topological groups (additional info and facts about topological groups) is often required to be continuous. The exponential map also yields a group homomorphism from the group of complex number (A number of the form a+bi where a and b are real numbers and i is the square root of -1) s C with addition to the group of non-zero complex numbers C G is a group homomorphism, we call it an endomorphism of G. www.absoluteastronomy.com /encyclopedia/g/gr/group_homomorphism.htm   (840 words)

 Station Information - Homomorphism   (Site not responding. Last check: 2007-11-07) A homomorphism, (or sometimes simply morphism) from one mathematical object to another of the same kind, is a mapping that is compatible with all relevant structure. A homomorphism which is also a bijection such that its inverse is also a homomorphism is called an isomorphism; two isomorphic objects are completely indistinguishable as far as the structure in question is concerned. A homomorphism from a set to itself is called an endomorphism, and if it is also an isomorphism is called an automorphism. www.stationinformation.com /encyclopedia/h/ho/homomorphism_1.html   (434 words)

 PlanetMath: field homomorphism   (Site not responding. Last check: 2007-11-07) is a field homomorphism, in particular it is a ring homomorphism. Remark: For this reason the terms ``field homomorphism'' and ``field monomorphism'' are synonymous. This is version 2 of field homomorphism, born on 2003-08-29, modified 2003-09-04. planetmath.org /encyclopedia/FieldHomomorphism.html   (125 words)

 Group homomorphism - Wikipedia, the free encyclopedia This approach is especially prevalent in areas of group theory where automata play a role, since it accords better with the convention that automata read words from left to right. For example, a homomorphism of topological groups is often required to be continuous. If the homomorphism h is a bijection, then one can show that its inverse is also a group homomorphism, and h is called a group isomorphism; in this case, the groups G and H are called isomorphic: they differ only in the notation of their elements and are identical for all practical purposes. en.wikipedia.org /wiki/Group_homomorphism   (846 words)

 Encyclopedia topic: Graph homomorphism   (Site not responding. Last check: 2007-11-07) In the mathematical (additional info and facts about mathematical) field of graph theory (additional info and facts about graph theory) a graph homomorphism is a mapping between two graphs (A drawing illustrating the relations between certain quantities plotted with reference to a set of axes) that respects their structure. If the homomorphism is a bijection (additional info and facts about bijection), then the inverse function is also a graph homomorphism, so is a graph isomorphism ((biology) similarity or identity of form or shape or structure). Graph homomorphism preserve connectedness (A relation between things or events (as in the case of one causing the other or sharing features with it)) and diconnectedness (additional info and facts about diconnectedness). www.absoluteastronomy.com /encyclopedia/g/gr/graph_homomorphism.htm   (432 words)

 Homomorphism   (Site not responding. Last check: 2007-11-07) it is verified that is a homomorphism of algebras; in particular... Then, it is clear that a homomorphism q of c into a subset [r.sup.*] of the... In this setting, a homomorphism is a map between two algebraic structures of the same type such that hallencyclopedia.com /Homomorphism   (762 words)

 Kernel (algebra) : Kernel of a homomorphism In mathematics, especially abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. In many cases, the kernel of a homomorphism is a subset of the domain of the homomorphism (specifically, those elements which are mapped to the identity element in the codomain). The group homomorphism f is injective iff the kernel of f consists of the identity element of G only. www.wordlookup.net /ke/kernel-of-a-homomorphism.html   (1008 words)

 PlanetMath: types of homomorphisms   (Site not responding. Last check: 2007-11-07) One can define similar natural concepts of homomorphisms for other algebraic structures, giving us ring homomorphisms, module homomorphisms, and a host of others. If the domain of a homomorphism is the same as its codomain (e.g. This is version 4 of types of homomorphisms, born on 2001-11-24, modified 2002-10-25. planetmath.org /encyclopedia/Homomorphism3.html   (289 words)

 SIGGS Theory Model: Homomorphism   (Site not responding. Last check: 2007-11-07) Homomorphism is a measure of similarity of connections. A classroom with low homomorphism would be one where students work independently, use different sources of information, and rely on the teacher less for direct instruction. If educational system homomorphism at time 2 is greater than homomorphism at time 1, then toput is nearly maximum and size degeneration is nearly maximum and complexity degeneration is nearly maximum. www.indiana.edu /~tedfrick/homomorp.html   (172 words)

 For a mapping F to be a homomorphism, it must be true that F(x o y) = F(x) * F(y) for the given operations o and * For a mapping F to be a homomorphism, it must be true that F(x o y) = F(x) * F(y) for the given operations o and *. For a mapping F to be an epimorphism, F must be a homomorphism that is surjective (onto). Let F be a homomorphism from the group G to the group G’. students.uww.edu /muellerbt15/Homomorphisms.htm   (608 words)

 Homomorphism - Wikipedia An example of homomorphism is given by group homomorphism. A homomorphism which is also a bijection is called an isomorphism; two isomorphic objects are completely indistinguishable as far as the structure in question is concerned. Any homomorphism f : X -> Y defines an equivalence relation on X by a ~ b iff f(a) = f(b). nostalgia.wikipedia.org /wiki/Homomorphism   (283 words)

 PlanetMath: dual homomorphism   (Site not responding. Last check: 2007-11-07) Furthermore, the dual of the identity homomorphism is the identity homomorphism of the dual space. Thus, using the language of category theory, the dualizing operation can be characterized as the homomorphism action of the contravariant, dual-space functor. This is version 6 of dual homomorphism, born on 2002-02-26, modified 2004-04-13. planetmath.org /encyclopedia/Adjoint2.html   (380 words)

 Homomorphisms Then we let pi be a homomorphism from J onto B + C, and f a homomorphism from J onto B. The universal property supplies a morphism psifrom B + C to B such that pi * psi= f (i.e., f(x) = psi(pi(x)) for all x). More precisely, if phi is a homomorphism A to B of abelian varieties, then the kernel of phi is usually not an abelian variety because it is rarely connected. The connected component C of ker (phi) and a morphism from C to the domain of phi where phi is a homomorphism of abelian varieties. www.math.lsu.edu /magma/text1323.htm   (3250 words)

 [No title]   (Site not responding. Last check: 2007-11-07) A homomorphism f from a group G to a group Gbar is a mapping from G into Gbar that preserves the group operation; that is, f(ab) = f(a)f(b) for all a,b in G. Page 194 Definition: Kernel of a Homomorphism. Let f be a homomorphism from a group G to a group Gbar and let g be an element of G. Then: (1) The identity is mapped to the identity, (2) Powers are preserved. Let f be a homomorphism from a group G to a group Gbar and let H be a subgroup of G. Then (1) f(H) is a subgroup (2) The image of a cyclic group is cyclic. orion.math.iastate.edu /hentzel/class.301.03/Dec.01   (327 words)

 HOMOMORPHISM   (Site not responding. Last check: 2007-11-07) A many-to-one mapping in effect representing (see representation) a pattern in the domain of the mapping by a simpler pattern in its range. Homomorphisms are important in establishing whether one system is a model of another and which properties of the original system the model retains. The inverse of a homomorphism is not a mapping. pespmc1.vub.ac.be /ASC/HOMOMORPHIS.html   (76 words)

 "Homomorphism in learning and teaching" We may say that processes of learning (in the sense 'learning by exploring') and teaching are homomorphic if for each stage of the former there is a correspondent stage in the latter. Teaching which is homomorphic to real learning process gives the teacher only those roles which in real process belong to the reality and to background knowledge. Homomorphism consists in the fact that two processes have exactly the same number of stages. www.leeds.ac.uk /educol/documents/000000835.htm   (2793 words)

 Pointers in creating a homomorphism Finding a homomorphism between two groups is often difficult especially for a first course in group theory. The set of homomorphisms between groups G and H is known as Hom(G,H), or in the case H=G, End(G). Another application shows that there are no non trivial homomorphisms from a cyclic group of order m to one of order n whenever m and n are coprime. www.math.csusb.edu /notes/advanced/algebra/gp/node17.html   (338 words)

 Definition and Elementary Properties In words, a homomorphism is just a map from one group to another which preserves the operation. This example shows that a factor group of a group is always a homomorphic image of the group. We note that the significance of this theorem is that it relates two seemingly unrelated concepts, i.e., concepts of factor group and homomorphic image. web.usna.navy.mil /~wdj/tonybook/gpthry/node37.html   (270 words)

 Element Operations   (Site not responding. Last check: 2007-11-07) Given a homomorphism a belonging to a submodule of Hom(M, N), and a homomorphism b belonging to a submodule of Hom(N, P), return the composition of the homomorphisms a and b as an element of Hom(M, P). Given a homomorphism a belonging to a submodule of Hom(M, N) with M and N having the same dimension, return the inverse of a as an element of Hom(N, M). The image of the homomorphism a belonging to the module H =Hom(M, N), returned as a submodule of N. Note that if the domain and codomain of a are matrix modules themselves, the image will be with respect to the appropriate action (right or left). www.math.uga.edu /~matthews/DOCS/MAGMA/text490.html   (571 words)

 The kernel of a homomorphism is the set of all elements in the domain that map to the identity element in the co-domain The kernel of a homomorphism is the set of all elements in the domain that map to the identity element in the co-domain. In other words, if F is a homomorphism from the group G to the group G’, the kernel, denoted Ker F, is the set of all elements x in F such that F(x)=e’ where e’ is the identity of G’. Since F is a homomorphism that is a bijection (injective and surjective), F can be classified as an isomorphism. students.uww.edu /muellerbt15/Kernel.htm   (476 words)

 Graph homomorphism - Encyclopedia, History, Geography and Biography In the mathematical field of graph theory a graph homomorphism is a mapping between two graphs that respects their structure. If the homomorphism $f$ is a bijection, then the inverse function is also a graph homomorphism, so $f$ is a graph isomorphism. Determining whether there is an isomorphism between two graphs is an important problem in computational complexity theory; see graph isomorphism problem. www.arikah.com /encyclopedia/Digraph_morphism   (410 words)

 Boolean algebra - Wikipedia, the free encyclopedia A truth assignment in propositional calculus is then a Boolean algebra homomorphism from this algebra to {0,1}. A homomorphism between the Boolean algebras A and B is a function f : A → B such that for all a, b in A: An isomorphism from A to B is a homomorphism from A to B which is bijective. www.wikipedia.org /wiki/Boolean_homomorphism   (2022 words)

 ABSTRACT ALGEBRA: OnLine Study Guide, Section 3.7 This is a group homomorphism from Z to G. If G is abelian, with its operation denoted additively, then we define µ : Z -> G by µ(n) = na. There are many important examples of group homomorphisms that are not isomorphisms, and, in fact, homomorphisms provide the way to relate one group to another. Examples 3.7.4 and 3.7.5 are important, because they give a complete description of all group homomorphisms between two cyclic groups. www.math.niu.edu /~beachy/abstract_algebra/study_guide/37.html   (626 words)

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