Homomorphism

In mathematics, a homomorphism between two objects is a structure-preserving map. This means that any relation holding between elements of the first structure must also hold between the mapped elements in the second structure. More intuitively, a homomorphism is a "relabeling" of an object that doesn't necessarily assign different labels to different elements.

A particular application of homomorphisms to polytopes is seen in the construction of step prismatic symmetry groups.

If a homomorphism is bijective, it's called an isomorphism.

Examples

 * Consider the function $$\varphi:\mathbb Z\to\mathbb Z_2$$ such that $$\varphi(n)=0$$ for even $$n$$, and $$\varphi(n)=1$$ for odd $$n$$. This function is a homomorphism between the groups $$(\mathbb Z,+)$$ and $$(\mathbb Z_2,+_2)$$. This is structure-preserving, since $$\varphi(a+b)=\varphi(a)+_2\varphi(b)$$ for any $$a,b\in\mathbb Z$$.
 * The function $$\varphi:\mathbb R\to\mathbb Z$$ such that $$\varphi(x)=\lfloor x\rfloor$$ is a homomorphism between the ordered sets $$(\mathbb R,\le_{\mathbb R})$$ and $$(\mathbb Z,\le_{\mathbb Z})$$. This is structure-preserving, since $$x\le_{\mathbb R} y$$ implies $$\varphi(x)\le_{\mathbb Z}\varphi(y)$$ for any $$x,y\in\mathbb R$$.
 * A polytope is orientable whenever there exists a homomorphism from its flags into those of the dyad, where the structure that's preserved is flag-adjacency.