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This article claims that a domain is a dcpo with a bottom element.
However, the book Continuous Lattices and Domains by G. GIERZ, K. H. HOFMANN, K. KEIMEL, J. D. LAWSON, M. MISLOVE, D. S. SCOTT claims (Definition I-1.6) that a domain is a dcpo that is also a continuous poset, i.e., a poset $L$ such that for each element $x\in L$ the set of elements well below $x$ is directed and its supremum equals $x$.
Which definition is correct?
The usual practice of domain theory papers and talks is to start by defining “domain” for the purposes of the paper or talk. It is a reusable word. Usually only compound words using the word “domain” have a fixed meaning, such as Scott domain (bounded complete algebraic dcpo), continuous Scott domain (bounded complete continuous dcpo), FS domain, SFP domain, L-domain. I wouldn’t use the plain word “domain” in a paper without first explicitly defining it, as it has been used with so many different meanings in the domain theory literature. Also the terminological conventions vary a bit depending on whether domain theory is used for the purposes of computation (e.g. programming language semantics, and theory of computability) or topology and algebra. For example, in programming language semantics papers one often encounters posets that have sups of ascending sequences, rather than directed sets, and a least element, because all the authors are interested in is the existence of (least) fixed point of continuous endo-functions, and these assumptions are enough for this purpose. Such posets are often called domains in such papers. But for applications of domain theory to topology, directed completeness is what one needs in general, and often we have more (even all sups - for example, a topological space is exponentiable if its lattice of open sets is a continuous dcpo - but this dcpo has all sups and moreover is a distributive lattice, and the continuous distributive lattices are precisely the topologies of exponentiable spaces, up to isomorphism). Because these applications and communities are so diverse, it is natural to see a divergence of terminology, even if many of the techniques are the same. So I think the nlab should refrain from defining “domain” .
(A little bit of this also happens in topology. For example, some papers and books say “all spaces are Hausdorff” in their preliminaries. Others don’t. For some authors “comp(act” means what other authors call “compact Hausdorff” (with quasi-compact reserved for the Heine-Borel property without the assumption of the Hausdorff separation axiom). But this terminological deviations are more controlled in topology than in domain theory, because people are aware of these specific deviations.)
Also, the nLab article on domain theory page linked from this page is historically inaccurate. It is not completely correct to say that “Domain theory has its origin in the problem in finding a viable denotational semantics for certain theories of computability”. The book cited above in these comments is a proof of this. Dana Scott proposed what he termed continuous lattices for denotational semantics. It turned out that the very same objects had made an appearance in topological algebra (where they were called Lawson semilattices). Out of the 6 authors of the above book and its previous edition called “A compendium of continuous lattices”, 5 where interested in the theory because of its connections with topology and algebra, and knew nothing about programming language semantics or computability, and 1 (Dana Scott) because of its connections to programming language semantics and to computability theory. Many years later, some of the 5 (Keimel, Lawson and Mislove) also made contributions motivated by computational problems. It was an amazing coincidence that the same mathematical structures arose in computation and in topology and algebra. This triggered a lot of activity, which eventually culminated in the book “continuous lattices” by these 6 authors, and its second edition “continuous lattices and domains”.
Following discussion here, I have moved one reference into this entry, which an Anonymous editor has suggested:
But it would be good to indicate what in that article the reader interested in directed-complete posets should take note of.
While I was at it, I have given the entry here a floating context menu, and I have touched the first Idea-sentence.
added pointer to:
(this may deserve its own entry, eventually, but for the moment I’ll keep it here)
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