Homomorphic and Isomorphic maps

A homomorphism is a function ${f}$ between two sets ${A}$ and ${B}$ such that the Group structure of the set is preserved.

Let the set ${A}$ have a law of composition ${+}$ such that forms a Group ${+:A\times A\rightarrow A}$ . Let also the set ${B}$ have a law of composition ${+:B\times B\rightarrow B}$

Definition: Let ${a,b\in A}$

$\begin{equation*} f(a+b)=f(a) + f(b) \end{equation*}$

Then ${f}$ is a homomorphic map.

It is important to note that the first ${+}$ is the law of composition on the set ${A}$ and the second ${+}$ is the law of composition on the set ${B}$ .

If ${x\in \mathbb{R}}$ and ${y\in \mathbb{R}}$ and ${f(x)=3x}$ . If ${(2,3)\in \mathbb{R}\times \mathbb{R}}$ then under ${+}$ , ${(2,3)\mapsto 5}$ . Then under ${f}$ , ${5\mapsto 3 \times 5 = 15}$ . However ${2\mapsto 6}$ and ${3\mapsto 9}$ then ${f(2) + f(3) = 15}$ . Hence we see that ${f(a+b)= f(a)+f(b)}$ . We therefore say that ${F}$ is a homomorphic map.

This map preserves structure when mapping from one set to another.

There can be an issue with a homomorphic map where the structure appears to collapse. Consider the set ${B}$ with an identity element ${e}$ . Let every ${x\in A}$ map to ${e\in B}$ under a map ${f}$ . Then ${a\mapsto e}$ and ${b\mapsto e}$ , but also ${a+b\mapsto e}$ . So ${f(a+b)=e}$ and ${f(a) + f(b) = e+e = e}$ . All elements map to the identity element. This appears to have lost some element of structure. The issues is that there is no inverse map from ${B}$ to ${A}$ that is a homomorphism. To have an inverse is to say that ${f}$ is a bijective map.

Definition: If ${f}$ is a homomorphic map and is also a bijective map then f is an isomorphic map.

This preserves structure. If two sets have an laws of composition and there exists an isomorphic map between them then they behave identically with respect to the law of composition(s).

Isomorphic maps are crucial in vector spaces as once a basis for the vector space is chosen, the vectors in that space are isomorphic to a set of real numbers. It is here from set theory to map / functions to isomorphism that we start to model physical actions!