Talk:Complete partial order

In the introduction, dcpo and cpo seem to stand for directed complete partial orders and complete partial orders, respectively, and seem to be distinct notions. In the definition section directed complete partial orders is defined, and is said to be abbreviated by both dcpo and (less commonly) cpo. Is it me, or is complete partial orders nowhere in the article defined? 145.97.197.215 (talk) 21:50, 18 August 2011 (UTC)

Hi. I think "complete partial order" or "cpo" is a vague term that can be used to mean either "directed complete partial order" or "omega complete partial order". That's my understanding. ComputScientist (talk) 15:58, 28 August 2011 (UTC)

Is this page about w-cpos?

Hi. Since the page is titled "complete partial order", which can mean both dcpo and w-cpo, why not make the article about both dcpos and w-cpos? I know there's another page about chain completeness, but I think that was supposed to be about having limits of all chains not just w-chains. ComputScientist (talk) 19:02, 30 August 2012 (UTC)

What kind of cpo?

The bit on Scott continuity says "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." Perhaps it should say "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 an $\omega$-cpo whenever Q is an $\omega$-cpo."

I was confused by this as well. My guess is that the original author meant that [P → Q] has a least element if Q has a least element. I've rewritten the sentence, using the terminology of "pointed dcpo" adopted in the article. Noamz (talk) 22:38, 1 September 2024 (UTC)

Dubious Example Partial Functions

The following example appears to be wrong:

The set of all partial functions on some given set S can be ordered by defining f ≤ g for functions f and 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 both functions are 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 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 specialization order of any sober space is a dcpo.

I think this only holds for finite domains.

As a counter example: Consider S to be the set of natural numbers. In the order constructed as above, the limited identity functions I_n (mapping each natural number less than or equal to n to itself) are ordered, i.e. I_n ≤ I_(n+1). The set {I_n} is a directed subset of the partial functions over S, but it has no supremum. — Preceding unsigned comment added by 91.66.22.113 (talk) 14:51, 8 January 2017 (UTC)

In your example, why do you think that the identity on S is not the supremum? Are you confused by "partial functions" meaning "not-necessarily-total functions", rather than "non-total functions"? 46.135.254.251 (talk) 22:46, 7 February 2017 (UTC)

"Closed preordered set" listed at Redirects for discussion

The redirect Closed preordered set has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2026 January 9 § Closed preordered set until a consensus is reached. –DMartin (talk) 21:00, 9 January 2026 (UTC)

Suggested simplifications

ω-completeness is just a special case of κ-completeness for an ordinal κ. I suggest this concept be moved to a separate article on κ-complete partial orders (or maybe a later section of this article), and focus this article mainly on completeness without regard for any size limit.

A complete partial order is the same thing as a complete lattice, for which an article already exists, so a link in this article to that article would dispose of that concept.

The reference to chain-complete posets in the second sentence of the Wikipedia article Bourbaki-Witt theorem links to this article. Yesterday (Saturday March 28) TakuyaMurata asked for "clarification".

What this article needs is more information about chain-complete posets. Since the first sentence of Domain theory#Important results says, "A poset D is a dcpo if and only if each chain in D has a supremum. (The 'if' direction relies on the axiom of choice.)", that's somewhat more information. So until the present article improves, I'll replace the link to this article with a link to the Important results section of the Domain theory article in the hope that this satisfies Taku.

Note that although the categories of continuous dcpo's and chain-complete partial orders have the same objects, the former has fewer morphisms than the latter, which is why Scott found it more useful for denotational semantics. This distinction is completely irrelevant to the Bourbaki-Witt theorem. Vaughan Pratt (talk) 22:12, 29 March 2026 (UTC)