In this thesis we undertake the study of the categories CCP(,o)-G(, )and CP(,o)-G, of chain-complete posets and continuous posets,(, )respectively, with Scott-continuous Galois-connections as their morphisms in both cases, inspired by the work of Dana S. Scott (see {Sc-72}) In Chapter 0 we establish definitions, theorems and terminology needed for the rest of the thesis. We initiate, in Section 1 of Chapter I, a systematic study of chain-complete posets and continuous posets having Scott-continuous Galois-connections as their morphisms. A main structural result is Theorem 1.17. In Section 2 of Chapter I, we study the function spaces of these objects, obtaining partial results in the case of continuous posets. In Section 3, we establish the existence of inverse limits of chain-complete posets, and we lay the mathematical foundation of denotational semantics of programming languages using chain-complete posets as ground objects. In Section 4 we characterize profinite posets, which arise in the semantics of parallel programming In Chapter II we develop several dualities for the categories introduced in Chapter I, using the Lawson duality between continuous posets and completely distributive lattices In Chapter III, we establish the existence of fixed point functors^in the category CCP(,o)-G and give two examples of these functors.(, )^We use these examples to initiate the study of topological combinatory algebras. We define the category TCA, of topological combinatory algebras, and proceed to study this category establishing several typical closedness properties. The category of continuous lattices provided the first examples topological combinatory algebras, using the existence in this category of topological spaces which are isomorphic to their own space of self-functions. Topological combinatory algebra D with D (TURNEQ) {D (--->) D}, are such that every self-function f can be obtained by application, i.e., there is x(,f) (ELEM) D such that f(y) = x(,f)(y) for every y (ELEM) D. For any poset X and topological combinatory algebra D, we form the topological combinatory algebra {X (--->) D} of Scott-continuous functions between X and D and we show that there are self-functions in these algebras which cannot be obtained by application / acase@tulane.edu
| Identifer | oai:union.ndltd.org:TULANE/oai:http://digitallibrary.tulane.edu/:tulane_25571 |
| Date | January 1981 |
| Contributors | Nino-Salcedo, Jaime (Author) |
| Publisher | Tulane University |
| Source Sets | Tulane University |
| Language | English |
| Detected Language | English |
| Rights | Access requires a license to the Dissertations and Theses (ProQuest) database., Copyright is in accordance with U.S. Copyright law |
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