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Author Archives: jgl
Ideal models II
Last time, we have seen that every completely metrizable space X has an ideal model, that is, that X can be embedded into an ideal domain Y in such a way that we can equate X with the subspace of … Continue reading
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Happy New Year 2016!
I had not posted a crossword puzzle for a long time, so here is one at last: in pdf format, or in AcrossLite format, as usual. Happy New Year!
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Ideal domains I
A few months ago, Keye Martin drew my attention to his results on so-called ideal models of spaces [1]. Ideal domains are incredibly specific dcpos: they are defined as dcpos where each non-finite element is maximal. Despite this, Keye Martin … Continue reading
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Domains XII
In my last post, I said I would have trouble finding time to write anything up in August, and sadly, this came out true. Late August, I went to the Domains XII conference, and it may be a good idea if I … Continue reading
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Adjoint Functor Theorems: GAFT and SAFT
You have probably sweated a lot at trying to understand the constructions of Part IV. They rest on a lot of topology and domain theory. Perhaps surprisingly (if you do not know it already), they are completely generic, and work in … Continue reading
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Powerdomains and hyperspaces IV: theories
Last time, we concluded with a mysterious observation. There is a theory, that of unital inflationary topological semi-lattices, which plays a fundamental role in the study of the Hoare powerspace. On the one hand, H(X) is the free sober such thing. On the … Continue reading
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Powerdomains and hyperspaces III: the theory of H
The last post was late. Let me compensate by being early this time. I had promised you that we would see why the theory of the Hoare powerspace monad was given by a small family of axioms, those of unital … Continue reading
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Powerdomains and hyperspaces II: monads
Let us deepen our understanding of the Hoare powerspace construction. We shall see that it defines a so-called monad. There would be many, many things to say about monads! I will only give a very superficial introduction here, trying to convince … Continue reading
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Hyperspaces and powerdomains I: closed and open sets
While a topological space is a space of points, a hyperspace is a space of subsets, with a suitable topology. Examples abound in the literature. For example, the so-called Smyth powerdomain (Proposition 8.3.25) is one. To start the series, let … Continue reading
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Iwamura’s Lemma, Markowsky’s Theorem, and ordinals
On p.61 of the book, there is a remark that the dcpos are exactly the chain-complete posets. This is a theorem by George Markowsky (1976). It is time I explained seriously how this worked. The first step is Iwamura’s Lemma … Continue reading
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