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Certainly Homomorphism is a prerequisite to establish an “Isomorphism”(Bijection), but what does a Homomorphism tell independently when it is established between two sets?

Homomorphism relates two sets as it is defined. But does it tell anything else? Or it is a tool for relating two sets only.

It would be nice to have an example where Homomorophism plays a big role besides being a condition for Isomorphism?

Michael
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  • Homomorphisms are as essential to group theory and ring theory as continuous functions are to topology. – Arthur Jan 18 '16 at 14:05
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    A homomorphism preserves operation, in order words preserves the structure from one set to another. It plays a similar or analogous role of continuous functions in Topology and rigid movements in Geometry. – Aaron Maroja Jan 18 '16 at 14:05
  • "Homo = the same" and "morph = form" When $f: A\to B$, the word homomorphism apply if $A$ having a structure $f(A)$ conserve this structure as subset of $B$ – Piquito Jan 18 '16 at 14:05
  • @AaronMaroja , "A homomorphism preserves operation" sounds good, is it like"if I can do addition in A, I can perform same thing in B" and this implies Both A, B has a "common property" (roughly speaking)? – Michael Jan 18 '16 at 14:16
  • @Jim An example you could work to visualize this is the homomorphism of $(\mathbb R, +)$ the addition group of real numbers and $(S^1, \cdot)$ the unitary circle with product, through the map $ x \mapsto e^{2\pi i x}$. Ir preverves operation meaning, $\phi (a + b) = \phi (a) \cdot \phi (b)$. – Aaron Maroja Jan 18 '16 at 14:22
  • I'm okay with "homomorphism preserves operation". However, I'm not okay with "homomorphism preserves structure". Saying a trivial function preserves structure is not a good thing, imo. – Aloizio Macedo Jan 18 '16 at 14:23
  • @AloizioMacedo , structure preserving is for Isomorphism, right? – Michael Jan 18 '16 at 14:25
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    It does preserve "some" of the structure though, see this answer and also this answer. – Aaron Maroja Jan 18 '16 at 14:26
  • Are you studying group theory? Wait a few weeks, you will some to the (first, second, and third) homomorphism theorems. – GEdgar Jan 18 '16 at 14:27
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    @AaronMaroja That is a philosophical aspect, a point of view, not a mathematical fact. To me, saying that a homomorphism preserves structure is the same thing as saying that if I took all the copper beams from a skyscraper and made 3 schools, I would have "preserved the structure" of the skyscraper. Saying preserve "some structure" is okay, though, and completely different from "preserving the structure". Again, this is a philosophical aspect. Nonetheless, the measure of "exactly how much" structure is preserved by a homomorphism is given by the isomorphism theorem. – Aloizio Macedo Jan 18 '16 at 14:48
  • @AloizioMacedo yes, you're right. – Aaron Maroja Jan 18 '16 at 14:58

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You are given two sets $A$ and $B$, both provided with a binary operation $*\>$. This means that in $A$ as well as in $B$ for certain triples $x$, $y$, $z$ it is true that $z=x*y\>$; e.g., $13=5+8$, or $91=7\cdot 13$. A map $\phi:\>A\to B$ is a homomorphism if it preserves such "incidences": $$z=x*y\quad\Longrightarrow\quad \phi(z)=\phi(x)*\phi(y)\ .$$

Christian Blatter
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