Cartesian closed categories of domains. (English) Zbl 0719.06004
CWI Tracts, 66. Amsterdam: Centrum voor Wiskunde en Informatica. 110 p. Dfl. 38.50 (1989).
This approximately 100 page thesis contains, in a very readable form, just about all one wants to know about the theory of domains, algebraic or continuous directed complete posets. Starting from the definition of a partial ordering, the author quickly gets to new results in domain theory. The work is divided into four chapters: Basic Concepts, Domains with least elements, Domains without a least element, and Continuous domains. The theme is to classify such domains by finding maximal subcategories with certain properties. The first such theorem is Smyth’s result that if D is an algebraic directed complete poset with least element and if the function space [D\(\to D]\) is \(\omega\)-algebraic, then D is bifinite. This theorem is generalized in the thesis in various ways: to the uncountable case, and to the case that the domains have no least element. The last chapter treats retracts of algebraic directed complete posets.
The paper is quite well-written and can serve as either an introduction to an interesting subject or as a source of conjectures and new results for the expert.
The paper is quite well-written and can serve as either an introduction to an interesting subject or as a source of conjectures and new results for the expert.
Reviewer: S.Bloom (Hoboken)
MSC:
06B35 | Continuous lattices and posets, applications |
68Q55 | Semantics in the theory of computing |
18D15 | Closed categories (closed monoidal and Cartesian closed categories, etc.) |
18B35 | Preorders, orders, domains and lattices (viewed as categories) |