|
| quote: | Originally posted by winston
The common theme is that a homomorphism is a function between two algebraic objects that respects the algebraic structure.
For example, a group is an algebraic object consisting of a set together with a single binary operation, satisfying certain axioms. If G and H are groups, a homomorphism from G to H is a function : G → H such that f(g_1 * g_2) = f(g_1) * f(g_2)\,\! for any elements g1, g2 ∈ G, where ∗ denotes the respective binary operations (the first ∗ denoting the operation in G, and the second ∗ denoting the operation in H).
When an algebraic structure includes more than one operation, homomorphisms are required to preserve each operation. For example, a ring possesses both addition and multiplication, and a homomorphism between two rings is a function such that
f(r+s) = f(r) + f(s)\qquad\text{and}\qquad f(rs) = f(r)\,f(s)
for any elements r and s of the domain ring. In most contexts, a homomorphism will map identity elements to identity elements, inverse elements to inverse elements, and so forth.
The notion of a homomorphism can be given a formal definition in the context of universal algebra, a field which studies ideas common to all algebraic structures. In this setting, a homomorphism : A → B is a function between two algebraic structures of the same type such that
f(\mu_A(a_1, \ldots, a_n)) = \mu_B(f(a_1), \ldots, f(a_n))\,
for each n-ary operation μ and for all elements a1,...,an ∈ A.
The real numbers are a ring, having both addition and multiplication. The set of all 2 Χ 2 matrices is also a ring, using matrix addition and matrix multiplication. Define a function between these rings by
f(r) = \begin{pmatrix} r & 0 \\ 0 & r \end{pmatrix}
where r is a real number. Then is a homomorphism of rings, since preserves both addition:
f(r+s) = \begin{pmatrix} r+s & 0 \\ 0 & r+s \end{pmatrix} = \begin{pmatrix} r & 0 \\ 0 & r \end{pmatrix} + \begin{pmatrix} s & 0 \\ 0 & s \end{pmatrix} = f(r) + f(s)
and multiplication:
f(rs) = \begin{pmatrix} rs & 0 \\ 0 & rs \end{pmatrix} = \begin{pmatrix} r & 0 \\ 0 & r \end{pmatrix} \begin{pmatrix} s & 0 \\ 0 & s \end{pmatrix} = f(r)\,f(s).
For another example, the nonzero complex numbers form a group under the operation of multiplication, as do the nonzero real numbers. (Zero must be excluded from both groups since it does not have a multiplicative inverse, which is required for elements of a group.) Define a function from the nonzero complex numbers to the nonzero real numbers by
f(z) = |z|.\,\!
That is, (z) is the absolute value (or modulus) of the complex number z. Then is a homomorphism of groups, since it preserves multiplication:
f(z_1 z_2) = |z_1 z_2| = |z_1|\,|z_2| = f(z_1)\,f(z_2).
Note that cannot be extended to a homomorphism of rings (from the complex numbers to the real numbers), since it does not preserve addition:
|z_1 + z_2| \ne |z_1| + |z_2|. |
Thank god someone's finally explained this to me in plain english. I don't know what I'd do without crackheads like winston explaining the world to me.
___________________
Captain Planet is gey.
Water, Fire, Earth, Wind, Heart???
These forces are supposed to combine to create Captain Planet?
Bullshit.
Those forces combine to create a soaking, boiling mudstorm on Valentine's Day.
|
