Intuition or Motivation behind definition of Homomorphism - Fraleigh p. 29

Question

p.29: A binary algebraic structure is a set $S$ together with a binary operation $*$ on $S$ and is denoted $<S, *>$

p.29: Let $<S,*>$ and $<S',*'>$ be binary algebraic structures.
An isomorphism of $S$ with $S'$ is a function $\phi$ mapping $S$ onto $S'$ such that
$1. \, \phi$ is one-to-one.
$2. \, \phi$ is onto.
$3. \, \phi$ satisfies the homomorphism property: $\phi(x*y) = \phi(x) *' \phi(y)$ for all $x, y \in S$.

What's the intuition or motivation behind the homomorphism property? Everything is from p. 29 Section 3 hence before groups, subgroups, ... are introduced.

I thought an isomorphism is a bijective linear transformation? How is a homomorphism related to a linear transformation : $T(a\mathbf{v} + b\mathbf{w}) = aT(\mathbf{v}) + bT(\mathbf{w})$, for all scalars $a,b$ and vectors $\mathbf{v,w}$ ?

Update Dec. 24, 2013 dezign wrote "a homomorphism is a function between sets which respects some sort of algebraic structure on the sets ". But how does this lead to $\phi(x*y) = \phi(x) *' \phi(y)$? Still looks weird. Wikipedia's paragraph on intuition doesn't explain it. It just does some examples.

Answer 1

In some approaches, such as Fraleigh's, it is natural to define isomorphisms before homomorphisms so that one can identify structures that look different but are really the same. This then leads to homomorphisms by relaxing the requirements of injectiveness and surjectiveness, leaving just the important property of preserving structure, in the sense of preserving corresponding operations.

A bijective linear transformation is an isomorphism of vector spaces. A homomorphism of vector spaces is just a linear transformation, which need be neither injective nor surjective.

Answer 2

In general, a homomorphism is a function between sets which respects some sort of algebraic structure on the sets (the appropriate language here is category theory, but we'll just stick to the basics). In the example you mentioned, a linear transformation respects the vector space structure of a vector space, i.e., you can perform the vector space operations before or after you apply the function, and you still get the same thing. Then this property may be generalized to a set with some sort of binary operation, i.e., the property that you can perform the binary operation before or after you apply the function and you still get the same thing. Functions with this property are called "homomorphisms".

Answer 3

The conditions 1) and 2) guarantee that $S$ and $S′$ are 'about the same' as sets. If they are satisfied then $S$ and $S′$ are isomorphic as sets. Conditions 1), 2) and 3) guarantee that $(S,⋆)$ and $(S′ ,⋆′ )$ are 'about the same' as binary algebraic structures. If they are satisfied then $(S,⋆)$ and $(S′ ,⋆′)$ are isomorphic as binary algebraic structures. You could say that 3) (homomorphism) takes care of the structure. I have added the tag 'categories'.