Editing Complete partial order (section)

== Examples ==

* Every finite poset is directed complete.
* All [[complete lattice]]s are also directed complete.
* For any poset, the set of all non-empty [[filter (mathematics)|filters]], ordered by [[inclusion (set theory)|subset inclusion]], is a dcpo. Together with the empty filter it is also pointed. If the order has binary [[join and meet|meets]], then this construction (including the empty filter) actually yields a [[complete lattice]].
* Every set ''S'' can be turned into a pointed dcpo by adding a least element ⊥ and introducing a flat order with ⊥&nbsp;≤&nbsp;''s'' and s&nbsp;≤&nbsp;''s'' for every ''s'' in ''S'' and no other order relations.
* The set of all [[partial function]]s on some given set ''S'' can be ordered by defining ''f''&nbsp;≤&nbsp;''g'' if and only if ''g'' extends ''f'', i.e. if the [[domain of a function|domain]] of ''f'' is a subset of the domain of ''g'' and the values of ''f'' and ''g'' agree on all inputs for which they are both defined. (Equivalently, ''f''&nbsp;≤&nbsp;''g'' if and only if ''f''&nbsp;⊆&nbsp;''g'' where ''f'' and ''g'' are identified with their respective [[graph of a function|graphs]].) This order is a pointed dcpo, where the least element is the nowhere-defined partial function (with empty domain). In fact, ≤ is also [[bounded complete]]. This example also demonstrates why it is not always natural to have a greatest element.
* The set of all [[linearly independent]] [[subset]]s of a [[vector space]] ''V'', ordered by [[inclusion (set theory)|inclusion]]. 
* The set of all partial [[choice function]]s on a collection of [[empty set|non-empty]] sets, ordered by restriction.
* The set of all [[prime ideal]]s of a [[ring (mathematics)|ring]], ordered by inclusion.
* The [[specialization order]] of any [[sober space]] is a dcpo.
* Let us use the term “[[deductive system]]” as a set of [[sentence (mathematical logic)|sentences]] closed under consequence (for defining notion of consequence, let us use e.g. [[Alfred Tarski]]'s algebraic approach<ref name=Tar-BizIg>Tarski, Alfred: Bizonyítás és igazság / Válogatott tanulmányok. Gondolat, Budapest, 1990. (Title means: Proof and truth / Selected papers.)</ref><ref name=BurSan-UnivAlg>[http://www.math.uwaterloo.ca/~snburris/index.html Stanley N. Burris] and H.P. Sankappanavar: [http://www.math.uwaterloo.ca/~snburris/htdocs/ualg.html A Course in Universal Algebra]</ref>). There are interesting theorems that concern a set of deductive systems being a directed-complete partial ordering.<ref name=seqdcpo>See online in p. 24 exercises 5–6 of §5 in [https://www.math.uwaterloo.ca/~snburris/htdocs/UALG/univ-algebra.pdf]. Or, on paper, see [[#_note-Tar-BizIg|Tar:BizIg]].</ref> Also, a set of deductive systems can be chosen to have a least element in a natural way (so that it can be also a pointed dcpo), because the set of all consequences of the empty set (i.e. “the set of the logically provable/logically valid sentences”) is (1) a deductive system (2) contained by all deductive systems.