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.

Definition
Say that a binary operation $$\cdot$$ is defined on two mathematical structures $$A$$ and $$B$$. We say that a function $$\varphi:A\to B$$ preserves the operation whenever
 * $$\varphi(x\cdot y)=\varphi(x)\cdot\varphi(y)$$ for all $$x,y\in A$$.

More generally, if a function $$f$$ with $$k$$ arguments is defined for both structures, we say that $$\varphi$$ preserves it whenever
 * $$\varphi(f(x_1,\ldots,x_k))=f(\varphi(x_1),\ldots,\varphi(x_k))$$ for all $$x_1,\ldots,x_k\in A$$.

We also consider preservation of a relation. If $$R$$ is a relation that takes $$k$$ arguments is defined on both $$A$$ and $$B$$, we say that $$\varphi$$ preserves it whenever
 * $$R(x_1,\ldots,x_k)\Rightarrow R(\varphi(x_1),\ldots,\varphi(x_k))$$ for all $$x_1,\ldots,x_k\in A$$.

Consider any type of mathematical structure, characterized by a set and various functions and relations on it, with any amounts of arguments (including 0). Examples of these include groups, partial orders, fields, and many others. A homomorphism is a function $$\varphi:A\to B$$ between any two instances of this structure such that all functions and relations on them are preserved.

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 $$(\mathbb Z_2,+)$$. This is structure-preserving, since $$\varphi(a+b)=\varphi(a)+\varphi(b)$$ 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 $$(\mathbb R,\le)$$ and $$(\mathbb Z,\le)$$. This is structure-preserving, since $$x\le y$$ implies $$\varphi(x)\le\varphi(y)$$ for any $$x,y\in\mathbb R$$.
 * There exists a surjective homomorphism between the omnitruncate of a polytope and any of its other truncates. The structure preserved is that of an abstract polytope.
 * A polytope is orientable whenever there exists a homomorphism from its flags into those of the dyad, where the structure that's preserved is the flag-adjacency relation.