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In mathematics, the phrase complete partial order is variously used to refer to at least three similar, but distinct, classes of
partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
s, characterized by particular completeness properties. Complete partial orders play a central role in
theoretical computer science computer science (TCS) is a subset of general computer science and mathematics that focuses on mathematical aspects of computer science such as the theory of computation, lambda calculus, and type theory. It is difficult to circumscribe the ...
: in denotational semantics and domain theory.


Definitions

A complete partial order, abbreviated cpo, can refer to any of the following concepts depending on context. * A partially ordered set is a directed-complete partial order (dcpo) if each of its directed subsets has a
supremum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lo ...
. A subset of a partial order is directed if it is non-empty 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 partially ordered set is a pointed directed-complete partial order if it is a dcpo with a least element. They are sometimes abbreviated cppos. * A partially ordered set is a ω-complete partial order (ω-cpo) if it is a poset in which every ω-chain (''x''1 ≤ ''x''2 ≤ ''x''3 ≤ ''x''4 ≤ ...) has a supremum that belongs to the poset. 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 Basis may refer to: Finance and accounting *Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates *Basis trading, a trading strategy consisting of ...
is also a dcpo (with the same basis). An ω-cpo (dcpo) with a basis is also called a continuous ω-cpo (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 Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...
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.


Examples

* Every finite poset is directed complete. * All complete lattices are also directed complete. * For any poset, the set of all non-empty filters, ordered by 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
partial function In mathematics, a partial function from a set to a set is a function from a subset of (possibly itself) to . The subset , that is, the domain of viewed as a function, is called the domain of definition of . If equals , that is, if is ...
s on some given set ''S'' can be ordered by defining ''f'' ≤ ''g'' if and only if ''g'' extends ''f'', i.e. if the 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'' ≤ ''g'' if and only if ''f'' ⊆ ''g'' where ''f'' and ''g'' are identified with their respective 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 In the mathematical field of order theory, a partially ordered set is bounded complete if all of its subsets that have some upper bound also have a least upper bound. Such a partial order can also be called consistently or coherently complete ( Viss ...
. This example also demonstrates why it is not always natural to have a greatest element. * The specialization order of any sober space is a dcpo. * Let us use the term “
deductive system A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A form ...
” as a set of sentences closed under consequence (for defining notion of consequence, let us use e.g.
Alfred Tarski Alfred Tarski (, born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician a ...
's algebraic approachTarski, Alfred: Bizonyítás és igazság / Válogatott tanulmányok. Gondolat, Budapest, 1990. (Title means: Proof and truth / Selected papers.)Stanley N. Burris
and H.P. Sankappanavar

/ref>). There are interesting theorems that concern a set of deductive systems being a directed-complete partial ordering.See online in p. 24 exercises 5–6 of §5 i

Or, on paper, see #_note-Tar-BizIg, Tar:BizIg.
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.


Properties

An ordered set ''P'' is a pointed dcpo if and only if every
chain A chain is a serial assembly of connected pieces, called links, typically made of metal, with an overall character similar to that of a rope in that it is flexible and curved in compression but linear, rigid, and load-bearing in tension. A ...
has a supremum in ''P'', i.e., ''P'' is chain-complete. Alternatively, an ordered set ''P'' is a pointed dcpo if and only if every order-preserving self-map of ''P'' has a least fixpoint.


Continuous functions and fixed-points

A
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
''f'' between two dcpos ''P'' and ''Q'' is called (Scott) continuous if 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 a monotone function. This notion of continuity is equivalent to the topological continuity induced by the
Scott topology Scott may refer to: Places Canada * Scott, Quebec, municipality in the Nouvelle-Beauce regional municipality in Quebec * Scott, Saskatchewan, a town in the Rural Municipality of Tramping Lake No. 380 * Rural Municipality of Scott No. 98, Sask ...
. The set of all continuous functions between two dcpos ''P'' and ''Q'' is denoted /nowiki>''P'' → ''Q''/nowiki>. 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 In category theory, a category is Cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in mat ...
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) *C ...
. Barendregt, Henk
''The lambda calculus, its syntax and semantics''
, North-Holland (1984)
Every order-preserving self-map ''f'' of a cpo (''P'', ⊥) has a least fixed-point.See Knaster–Tarski theorem; The foundations of program verification, 2nd edition, Jacques Loeckx and Kurt Sieber, John Wiley & Sons, , Chapter 4; the Knaster–Tarski theorem, formulated over cpo's, is given to prove as exercise 4.3-5 on page 90. If ''f'' is continuous then this fixed-point is equal to the supremum of the iterates (⊥, ''f'' (⊥), ''f'' (''f'' (⊥)), … ''f'' ''n''(⊥), …) of ⊥ (see also the
Kleene fixed-point theorem In the mathematical areas of order and lattice theory, the Kleene fixed-point theorem, named after American mathematician Stephen Cole Kleene, states the following: :Kleene Fixed-Point Theorem. Suppose (L, \sqsubseteq) is a directed-complete pa ...
).


See also

Directed completeness alone is quite a basic property that occurs often in other order-theoretic investigations, using for instance
algebraic poset {{Unreferenced, date=December 2008 In the mathematical area of order theory, the compact elements or finite elements of a partially ordered set are those elements that cannot be subsumed by a supremum of any non-empty directed set that does not alr ...
s and the
Scott topology Scott may refer to: Places Canada * Scott, Quebec, municipality in the Nouvelle-Beauce regional municipality in Quebec * Scott, Saskatchewan, a town in the Rural Municipality of Tramping Lake No. 380 * Rural Municipality of Scott No. 98, Sask ...
. Directed completeness relates in various ways to other completeness notions such as chain completeness.


Notes


References

* {{DEFAULTSORT:Complete Partial Order Order theory ru:Частично упорядоченное множество#Полное частично упорядоченное множество