Editing Isomorphism (section)

==Examples==

===Logarithm and exponential===
Let <math> \R ^+ </math> be the [[multiplicative group]] of [[positive real numbers]], and let <math>\R</math> be the additive group of real numbers.

The [[logarithm function]] <math>\log : \R^+ \to \R</math> satisfies <math>\log(xy) = \log x + \log y</math> for all <math>x, y \in \R^+,</math> so it is a [[group homomorphism]]. The [[exponential function]] <math>\exp : \R \to \R^+</math> satisfies <math>\exp(x+y) = (\exp x)(\exp y)</math> for all <math>x, y \in \R,</math> so it too is a homomorphism.

The identities <math>\log \exp x = x</math> and <math>\exp \log y = y</math> show that <math>\log</math> and <math>\exp </math> are [[inverse function|inverses]] of each other. Since <math>\log</math> is a homomorphism that has an inverse that is also a homomorphism, <math>\log</math> is an [[Group isomorphism|isomorphism of groups]], i.e., <math>\R^+ \cong \R</math> via the isomorphism <math>\log x</math>.

The <math>\log</math> function is an isomorphism which translates multiplication of positive real numbers into addition of real numbers. This facility makes it possible to multiply real numbers using a [[ruler]] and a [[table of logarithms]], or using a [[slide rule]] with a logarithmic scale.

===Integers modulo 6===
Consider the group <math>(\Z_6, +),</math> the integers from 0 to 5 with addition [[Modular arithmetic|modulo]]&nbsp;6. Also consider the group <math>\left(\Z_2 \times \Z_3, +\right),</math> the ordered pairs where the ''x'' coordinates can be 0 or 1, and the y coordinates can be 0, 1, or 2, where addition in the ''x''-coordinate is modulo 2 and addition in the ''y''-coordinate is modulo 3.

These structures are isomorphic under addition, under the following scheme:
<math display="block">\begin{alignat}{4}
(0, 0) &\mapsto 0 \\
(1, 1) &\mapsto 1 \\
(0, 2) &\mapsto 2 \\
(1, 0) &\mapsto 3 \\
(0, 1) &\mapsto 4 \\
(1, 2) &\mapsto 5 \\
\end{alignat}</math>
or in general <math>(a, b) \mapsto (3 a + 4 b) \mod 6.</math>

For example, <math>(1, 1) + (1, 0) = (0, 1),</math> which translates in the other system as <math>1 + 3 = 4.</math>

Even though these two groups "look" different in that the sets contain different elements, they are indeed '''isomorphic''': their structures are exactly the same. More generally, the [[direct product of groups|direct product]] of two [[cyclic group]]s <math>\Z_m</math> and <math>\Z_n</math> is isomorphic to <math>(\Z_{mn}, +)</math> if and only if ''m'' and ''n'' are [[coprime]], per the [[Chinese remainder theorem]].

===Relation-preserving isomorphism===
If one object consists of a set ''X'' with a [[binary relation]] R and the other object consists of a set ''Y'' with a binary relation S then an isomorphism from ''X'' to ''Y'' is a bijective function <math>f : X \to Y</math> such that:<ref>{{Cite book|author=Vinberg, Ėrnest Borisovich|title=A Course in Algebra|publisher=American Mathematical Society|year=2003|isbn=9780821834138|page=3|url=https://books.google.com/books?id=kd24d3mwaecC&pg=PA3}}</ref>
<math display="block">\operatorname{S}(f(u),f(v)) \quad \text{ if and only if } \quad \operatorname{R}(u,v) </math>

S is [[Reflexive relation|reflexive]], [[Irreflexive relation|irreflexive]], [[Symmetric relation|symmetric]], [[Antisymmetric relation|antisymmetric]], [[Asymmetric relation|asymmetric]], [[Transitive relation|transitive]], [[Connected relation|total]], [[Homogeneous relation#Properties|trichotomous]], a [[partial order]], [[total order]], [[well-order]], [[strict weak order]], [[Strict weak order#Total preorders|total preorder]] (weak order), an [[equivalence relation]], or a relation with any other special properties, if and only if R is.

For example, R is an [[Order theory|ordering]] ≤ and S an ordering <math>\scriptstyle \sqsubseteq,</math> then an isomorphism from ''X'' to ''Y'' is a bijective function <math>f : X \to Y</math> such that
<math display="block">f(u) \sqsubseteq f(v) \quad \text{ if and only if } \quad  u \leq v.</math>
Such an isomorphism is called an {{em|[[order isomorphism]]}} or (less commonly) an {{em|isotone isomorphism}}.

If <math>X = Y,</math> then this is a relation-preserving [[automorphism]].