Editing Preorder (section)

===Computer science===
In computer science, one can find examples of the following preorders.
* [[Big O notation|Asymptotic order]] causes a preorder over  functions <math>f: \mathbb{N} \to \mathbb{N}</math>. The corresponding equivalence relation is called [[Asymptotic_analysis#Definition|asymptotic equivalence]].
* [[Polynomial-time reduction|Polynomial-time]], [[Many-one reduction|many-one (mapping)]] and [[Turing reduction]]s are preorders on complexity classes.
* [[Subtyping]] relations are usually preorders.<ref>{{cite book |last=Pierce |first=Benjamin C. |author-link=Benjamin C. Pierce
 |date=2002 |title=Types and Programming Languages |title-link=Types and Programming Languages |location=Cambridge, Massachusetts/London, England |publisher=The MIT Press |pages=182ff |isbn=0-262-16209-1}}</ref>
* [[Simulation preorder]]s are preorders (hence the name).
* [[Reduction relation]]s in [[abstract rewriting system]]s.
* The [[encompassment preorder]] on the set of [[term (logic)|term]]s, defined by <math>s \lesssim t</math> if a [[term (logic)#Operations with terms|subterm]] of ''t'' is a [[substitution instance]] of ''s''.
* [[Theta-subsumption]],<ref>{{cite journal |last=Robinson | first=J. A. |title=A machine-oriented logic based on the resolution principle |journal=ACM |volume=12 |number=1 |pages=23–41 |year=1965 | doi=10.1145/321250.321253 | s2cid=14389185 |doi-access=free }}</ref> which is when the literals in a disjunctive first-order formula are contained by another, after applying a [[Substitution (logic)|substitution]] to the former.