Homomorphisms, Linear Transformations, and Distributivity

Question

What are the differences between homomorphisms, linear transformations, and distributive operations? To me, they all seem essentially the same, they just are different names for the same phenomenon based on context.

Answer 1

Homomorphism refers to mappings $f \,: X \to Y$ that preserve some kind of structure. Which structure that is in particular depends on the context, or must explicitly be specified. For example, if $f$ is a mapping from group $X$ to group $Y$, then saying "$f$ is an homomorphism" means that $f$ is compatible with the group structure, i.e. that $f(a\cdot b) = f(a)\cdot f(b)$. If $X$ and $Y$ were fields, it would mean that $f(a+b) = f(a) + f(b)$ as well as $f(ab) = f(a)f(b)$. You are correct that for the case of vector spaces, homomorphism means the same as linear.

Distributivity refers to a pair of two operations $u,v \,:\, X\times X \to X$. If $u(x,v(y,z)) = v(u(x,y),u(x,z))$ then $u$ distributes over $v$. Note that you're dealing with mappings of arity two here, instead of arity one as in the case of homomorphisms, and with mappings of a set $X$ to itself, instead of to a different set $Y$.

But yeah, for two operations $u,v$ where $(X,v)$ is a structure of type V (e.g. a group), saying that $x \mapsto u(v,x)$ is a $V$-homomorphism for every $x$ would be the same as saying $u$ distributes over $v$. In the case of a field $F$, that comes down to saying that multiplication with a fixed element $a$, i.e. the map $x \mapsto a\cdot x$, is a group homomorphism from $(F,+)$ to itself. But whether that way of expressing things would be an improvement is another question...

Answer 2

A homomorphism is typically a transformation which preserves some sort of structure in an algebraic object: $$\phi(a*b)=\phi(a)\cdot \phi(b)$$

One operation f is set to distribute over another g if $$f(c,g(a,b))=g(f(c,a),f(c,b))$$

A linear transformation (very different) is anything that distributes over both addition and scalar multiplication, usually in a vector space: $$L(c_1x_1+c_2x_2)=c_1L(x_1)+c_2L(x_2)$$