Complete partial order

Inmathematics,the phrasecomplete partial orderis variously used to refer to at least three similar, but distinct, classes ofpartially ordered sets,characterized by particularcompleteness properties.Complete partial orders play a central role intheoretical computer science:indenotational semanticsanddomain theory.

Definitions

The termcomplete partial order,abbreviatedcpo,has several possible meanings depending on context.

A partially ordered set is adirected-complete partial order(dcpo) if each of itsdirected subsetshas asupremum.(A subset of a partial order is directed if it isnon-emptyand every pair of elements has an upper bound in the subset.) In the literature, dcpos sometimes also appear under the labelup-complete poset.

Apointed directed-complete partial order(pointed dcpo,sometimes abbreviatedcppo), is a dcpo with aleast element(usually denoted$\bot$). Formulated differently, a pointed dcpo has a supremum for every directedor emptysubset. The termchain-complete partial orderis also used, because of the characterization of pointed dcpos as posets in which everychainhas a supremum.

A related notion is that ofω-complete partial order(ω-cpo). These are posets in which every ω-chain ($x_{1}\leq x_{2}\leq x_{3}\leq...$) has a supremum that belongs to the poset. The same notion can be extended to othercardinalitiesof chains.

Every dcpo is an ω-cpo, since every ω-chain is a directed set, but theconverseis not true. However, every ω-cpo with abasisis also a dcpo (with the same basis).An ω-cpo (dcpo) with a basis is also called acontinuousω-cpo (or continuous dcpo).

Note thatcomplete partial orderis never used to mean a poset in whichallsubsets have suprema; the terminologycomplete latticeis used for this concept.

Requiring the existence of directed suprema can be motivated by viewing directed sets as generalized approximation sequences and suprema aslimitsof the respective (approximative) computations. This intuition, in the context of denotational semantics, was the motivation behind the development ofdomain theory.

Thedualnotion of a directed-complete partial order is called afiltered-complete partial order.However, this concept occurs far less frequently in practice, since one usually can work on the dual order explicitly.

By analogy with theDedekind–MacNeille completionof a partially ordered set, every partially ordered set can be extended uniquely to a minimal dcpo.

Examples

• Every finite poset is directed complete.
• Allcomplete latticesare also directed complete.
• For any poset, the set of all non-emptyfilters,ordered bysubset inclusion,is a dcpo. Together with the empty filter it is also pointed. If the order has binarymeets,then this construction (including the empty filter) actually yields acomplete lattice.
• Every setScan be turned into a pointed dcpo by adding a least element ⊥ and introducing a flat order with ⊥ ≤sand s ≤sfor everysinSand no other order relations.
• The set of allpartial functionson some given setScan be ordered by definingf≤gif and only ifgextendsf,i.e. if thedomainoffis a subset of the domain ofgand the values offandgagree on all inputs for which they are both defined. (Equivalently,f≤gif and only iff⊆gwherefandgare identified with their respectivegraphs.) This order is a pointed dcpo, where the least element is the nowhere-defined partial function (with empty domain). In fact, ≤ is alsobounded complete.This example also demonstrates why it is not always natural to have a greatest element.
• The set of alllinearly independentsubsetsof avector spaceV,ordered byinclusion.
• The set of all partialchoice functionson a collection ofnon-emptysets, ordered by restriction.
• The set of allprime idealsof aring,ordered by inclusion.
• Thespecialization orderof anysober spaceis a dcpo.
• Let us use the term “deductive system”as a set ofsentencesclosed under consequence (for defining notion of consequence, let us use e.g.Alfred Tarski's algebraic approach). There are interesting theorems that concern a set of deductive systems being a directed-complete partial ordering.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.

Characterizations

An ordered set is a dcpo if and only if every non-emptychainhas 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 ischain-complete.Proofs rely on theaxiom of choice.

Alternatively, an ordered set$P$is a pointed dcpo if and only if everyorder-preservingself-map of$P$has a leastfixpoint.

Continuous functions and fixed-points

Afunctionfbetween two dcposPandQis called(Scott) continuousif it maps directed sets to directed sets while preserving their suprema:

• $f(D)\subseteq Q$is directed for every directed$D\subseteq P$.
• $f(\sup D)=\sup f(D)$for every directed$D\subseteq P$.

Note that every continuous function between dcpos is amonotone function. This notion of continuity is equivalent to thetopological continuityinduced by theScott topology.

The set of all continuous functions between two dcposPandQis denoted [P→Q]. Equipped with thepointwise order,this is again a dcpo, and pointed wheneverQis pointed. Thus the complete partial orders with Scott-continuous maps form acartesian closedcategory.

Every order-preserving self-mapfof a pointed dcpo (P,⊥) has a least fixed-point.Iffis continuous then this fixed-point is equal to the supremum of theiterates(⊥,f (⊥),f (f (⊥)),…fⁿ(⊥),…) of ⊥ (see also theKleene fixed-point theorem).

Another fixed point theorem is theBourbaki-Witt theorem,stating that if$f$is a function from a dcpo to itself with the property that$f(x)\geq x$for all$x$,then$f$has a fixed point. This theorem, in turn, can be used to prove that Zorn's lemma is a consequence of the axiom of choice.

See also

Directed completeness alone is quite a basic property that occurs often in other order-theoretic investigations, using for instancealgebraic posetsand theScott topology.

Directed completeness relates in various ways to othercompletenessnotions such aschain completeness.

References

• Davey, B.A.;Priestley, H. A.(2002).Introduction to Lattices and Order(Second ed.). Cambridge University Press.ISBN0-521-78451-4.