Editing Isomorphism (section)

===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]].