On continuous posets and their applications

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Title 
On continuous posets and their applications 
Author 
NinoSalcedo, Jaime 
School 
Tulane University 
Academic Field 
Mathematics 
Abstract 
In this thesis we undertake the study of the categories CCP(,o)G(, )and CP(,o)G, of chaincomplete posets and continuous posets,(, )respectively, with Scottcontinuous Galoisconnections as their morphisms in both cases, inspired by the work of Dana S. Scott (see {Sc72}) 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 chaincomplete posets and continuous posets having Scottcontinuous Galoisconnections 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 chaincomplete posets, and we lay the mathematical foundation of denotational semantics of programming languages using chaincomplete 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 selffunctions. Topological combinatory algebra D with D (TURNEQ) {D (>) D}, are such that every selffunction 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 Scottcontinuous functions between X and D and we show that there are selffunctions in these algebras which cannot be obtained by application 
Language 
eng 
Degree Date 
1981 
Degree 
Ph.D 
Publisher 
Tulane University 
Publication Date 
1981 
Source 
109 p., Dissertation Abstracts International, Volume: 4206, Section: B, page: 2404 
Identifier 
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Rights 
Copyright is in accordance with U.S. Copyright law 
Contact Information 
acase@tulane.edu 
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