# cpos * table of contents {: toc} ## Definition The abbreviation __cpo__ stands for "complete partial order", (beware that it does *not* generally refer to a [[partial order]] that is [[complete category|complete]] in the sense that the latter word is [[complete category|used]] in [[category theory]]: A partial order that is [[complete category|complete as a category]] is instead generally called a *[[complete lattice]]*.) Instead, a "cpo" generally refers to either a poset that has [[suprema]] for [[directed set|directed]] subsets (a.k.a. a __[[dcpo]]__) or for [[chains]] indexed by the [[ordinal]] $\omega$ (a.k.a. an **[[omega-cpo|$\omega$-cpo]]** --- a weaker condition). Note also that suprema are [[colimits]] in a poset rather than [[limits]], so this is actually a [[cocomplete category|cocompleteness]] condition rather than a completeness one. The [[morphisms]] between cpos are usually taken to be the the [[Scott-continuous function]]s ([[continuous functions]] with respect to the [[Scott topology]]). Cpos and Scott-continuous maps form a category $CPO$. ## Applications Cpos are important in [[domain theory]]. ## References * [[Jiri Adamek]], [[Horst Herrlich]], and [[George Strecker]], _Abstract and concrete categories: the joy of cats_. [free online](http://katmat.math.uni-bremen.de/acc/acc.pdf) [[!redirects cpo]] [[!redirects cpos]] [[!redirects CPO]]