Editing Complete partial order (section)

== Characterizations ==

An ordered set is a dcpo if and only if every non-empty [[chain (order theory)|chain]] has a supremum. As a corollary, an ordered set is a pointed dcpo if and only if every (possibly empty) chain has a supremum, i.e., if and only if it is [[chain-complete partial order|chain-complete]].<ref name="markowsky-1976"/><ref>
{{cite web
 | url = https://topology.lmf.cnrs.fr/iwamuras-lemma-kowalskys-theorem-and-ordinals/
 | title = Iwamura's Lemma, Markowsky's Theorem and ordinals
 | last = Goubault-Larrecq
 | first = Jean
 | date = February 23, 2015
 | access-date = January 6, 2024
}}</ref><ref>
{{cite book
 | last = Cohn
 |first = Paul Moritz
 | title = Universal Algebra
 | publisher = Harper and Row
 | page = 33
}}</ref><ref>
{{ cite web
 | url = https://topology.lmf.cnrs.fr/markowsky-or-cohn/
 | title = Markowsky or Cohn?
 | last = Goubault-Larrecq
 | first = Jean
 | date = January 28, 2018
 | access-date = January 6, 2024
}}</ref> Proofs rely on the [[axiom of choice]].

Alternatively, an ordered set <math>P</math> is a pointed dcpo if and only if every [[order-preserving]] self-map of <math>P</math> has a least [[fixpoint]].