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I’m not sure that the definition on this page is correct. Despite how wrong it seems to a category theorist, I think the adjective “complete” in “cpo” usually refers only to a countable sort of completeness. According to wikipedia, a “cpo” can mean at least three different things dependent on context, but “never” a partial order that’s actually complete as a category (i.e. a complete lattice).
I’ve never heard of cpos, but only $\omega$-cpos which are used to model recursion in simply typed lambda calculus.
I think people use “cpo” to mean either “$\omega$-cpo” or “dcpo”. I don’t think I’ve ever seen it used to mean complete lattice. The weird thing is that “cpo” doesn’t seem to appear in the given reference, the AHS book.
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