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# Domains

We will begin by reading the first five chapters of the *Pisa Notes* by Gordon Plotkin. The weekly meetings will consist of discussion about our different solutions/attempts to solve the exercises, in the fashion of
Moore.

### Errata

- Page 1, one should have
*C*:Stat→*C*. - Page 9 ex 8 is lacking an assumption: the dcpo should be pointed.
- Page 22 ex 12: the given definition of
*V-sequentiality*is wrong. It should be read:

“[…] iff eitherfis a constant or,for each, there is an integerxi(with 0≤i≤n) such that […]”

### Hints for some exercises

- 1.8. Pataria found a proof (in 1997!) that doesn't invoke transfinite induction (link, exercise O-2.21).
- 1.19. There is a tricky proof in the book
*Domains and lambda-calculi*by R. Amadio and P.L. Curien. It proceeds in two steps (proposition 7.1.27 and corollary 7.1.28):- There is no monotone surjection from
*D*to*D*→ 𝕆, hence there is no monotone surjection from^{op}*D*to (*D*→𝕆)→𝕆, - There is no retraction from
*D*→*D*to*D*since otherwise that would imply the above (using the retraction from*D*to 𝕆).

- There is no monotone surjection from
- The function space construction does not preserve ω-algebraicity. The counterexamples are lacking some explanation: the idea is to show that approx(id
_{D}) is not directed, hence*D → D*cannot be ω-algebraic.

Relevant links:

*Pisa Notes**LCF considered as a programming language**Domain-Theoretic Foundations of Functional Programming**Domain theory*(Abramsky and Jung)*Introduction to Domain Theory*by Graham Hutton (handwritten slides)

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