Complete partial order

In mathematics, the phrase complete partial order is variously used to refer to at least three similar, but distinct, classes of partially ordered sets, characterized by particular completeness properties. Complete partial orders play a central role in theoretical computer science: in denotational semantics and domain theory.

Definitions

The term complete partial order, abbreviated cpo, has several possible meanings depending on context.

A partially ordered set is a directed-complete partial order (dcpo) if each of its

and every pair of elements has an upper bound in the subset.) In the literature, dcpos sometimes also appear under the label up-complete poset.

A pointed directed-complete partial order (pointed dcpo, sometimes abbreviated cppo), is a dcpo with a

least element

(usually denoted ${\displaystyle \bot }$). Formulated differently, a pointed dcpo has a supremum for every directed or empty subset. The term chain-complete partial order is also used, because of the characterization of pointed dcpos as posets in which every

chain

has a supremum.

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

Every dcpo is an ω-cpo, since every ω-chain is a directed set, but the converse is not true. However, every ω-cpo with a basis is also a dcpo (with the same basis).^[2] An ω-cpo (dcpo) with a basis is also called a continuous ω-cpo (or continuous dcpo).

Note that complete partial order is never used to mean a poset in which all subsets have suprema; the terminology complete lattice is used for this concept.

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

The dual notion of a directed-complete partial order is called a filtered-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 the Dedekind–MacNeille completion of a partially ordered set, every partially ordered set can be extended uniquely to a minimal dcpo.^[1]

Examples

• Every finite poset is directed complete.
• All complete lattices are also directed complete.
• For any poset, the set of all non-empty
  subset inclusion, is a dcpo. Together with the empty filter it is also pointed. If the order has binary 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 ⊥ ≤ s and s ≤ s for every s in S and no other order relations.
• The set of all
  bounded complete
  . This example also demonstrates why it is not always natural to have a greatest element.
• The set of all
  inclusion
  .
• The set of all partial choice functions on a collection of non-empty sets, ordered by restriction.
• The set of all prime ideals of a 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 sentences closed under consequence (for defining notion of consequence, let us use e.g. Alfred Tarski's algebraic approach^[3][4]). There are interesting theorems that concern a set of deductive systems being a directed-complete partial ordering.^[5]
  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-empty

.

Alternatively, an ordered set ${\displaystyle P}$ is a pointed dcpo if and only if every

order-preserving

self-map of ${\displaystyle P}$ has a least

fixpoint

.

Continuous functions and fixed-points

A function f between two dcpos P and Q is called (Scott) continuous if it maps directed sets to directed sets while preserving their suprema:

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

Note that every continuous function between dcpos is a

. This notion of continuity is equivalent to the .

The set of all continuous functions between two dcpos P and Q is denoted [P → Q]. Equipped with the pointwise order, this is again a dcpo, and a cpo whenever Q is a cpo. Thus the complete partial orders with Scott-continuous maps form a cartesian closed category.^[9]

Every order-preserving self-map f of a cpo (P, ⊥) has a least fixed-point.^[10] If f is continuous then this fixed-point is equal to the supremum of the iterates (⊥, f (⊥), f (f (⊥)), … fⁿ(⊥), …) of ⊥ (see also the Kleene fixed-point theorem).

Another fixed point theorem is the

, stating that if ${\displaystyle f}$ is a function from a dcpo to itself with the property that ${\displaystyle f(x)\geq x}$ for all ${\displaystyle x}$, then ${\displaystyle f}$ 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.^[11][12]

See also

Directed completeness alone is quite a basic property that occurs often in other order-theoretic investigations, using for instance

.

Directed completeness relates in various ways to other

.

Notes

References

• Davey, B.A.;
  ISBN 0-521-78451-4
  .