Editing Homomorphism (section)

== Examples ==

[[File:Exponentiation as monoid homomorphism svg.svg|thumb|[[Monoid]] homomorphism <math>f</math> from the monoid {{math|{{color|#008000|('''N''', +, 0)}}}} to the monoid {{math|{{color|#800000|('''N''', ×, 1)}}}}, defined by <math>f(x) = 2^x</math>. It is [[Injective function|injective]], but not [[Surjective function|surjective]].]]
The [[real number]]s are a [[ring (mathematics)|ring]], having both addition and multiplication.  The set of all 2×2 [[matrix (mathematics)|matrices]] is also a ring, under [[matrix addition]] and [[matrix multiplication]].  If we define a function between these rings as follows:

<math display="block">f(r) = \begin{pmatrix}
   r & 0 \\
   0 & r
\end{pmatrix}</math>

where {{mvar|r}} is a real number, then {{mvar|f}} is a homomorphism of rings, since {{mvar|f}} preserves both addition:

<math display="block">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)</math>

and multiplication:

<math display="block">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).</math>

For another example, the nonzero [[complex number]]s form a [[group (mathematics)|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 <math>f</math> from the nonzero complex numbers to the nonzero real numbers by

<math display="block">f(z) = |z| .</math>

That is, <math>f</math> is the [[absolute value]] (or modulus) of the complex number <math>z</math>.  Then <math>f</math> is a homomorphism of groups, since it preserves multiplication:

<math display="block">f(z_1 z_2) = |z_1 z_2| = |z_1| |z_2| = f(z_1) f(z_2).</math>

Note that {{math|''f''}} cannot be extended to a homomorphism of rings (from the complex numbers to the real numbers), since it does not preserve addition:

<math display="block">|z_1 + z_2| \ne |z_1| + |z_2|.</math>

As another example, the diagram shows a [[monoid]] homomorphism <math>f</math> from the monoid <math>(\mathbb{N}, +, 0)</math> to the monoid <math>(\mathbb{N}, \times, 1)</math>. Due to the different names of corresponding operations, the structure preservation properties satisfied by <math>f</math> amount to <math>f(x+y) = f(x) \times f(y)</math> and <math>f(0) = 1</math>.

A [[composition algebra]] <math>A</math> over a field <math>F</math> has a [[quadratic form]], called a ''norm'', <math>N: A \to F</math>, which is a group homomorphism from the [[multiplicative group]] of <math>A</math> to the multiplicative group of <math>F</math>.