Dualities of domains

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Title 
Dualities of domains 
Author 
Zhang, Han 
School 
Tulane University 
Academic Field 
Mathematics Computer Science 
Abstract 
The basic mathematical structures used in denotational semantics to model programming languages are algebraic posets. A more general notion is that of a continuous poset. It is known that when equipped with the Scott topology, a continuous poset is locally compact and has a basis of open filters. It turns out that these two conditions are sufficient to ensure that a poset is continuous. In general, if a poset has only a basis of open filters, we obtain that the Scott open interior map from its lattice of upper sets to its lattice of open sets preserves finite union, i.e., $(A\ \cup\ B)\sp\circ = A\sp\circ\ \cup\ B\sp\circ$ if A and B are upper sets Spectral theory studies the relation between a topological space and its lattice of closed (or open) sets. It is known that when we apply this theory to those spaces which arise from the Scott topology of a continuous dcpo (directed complete poset), a duality between continuous dcpo's and completely distributive lattices is established and is known as the Lawson Duality. If we impose the condition that each continuous dcpo has a bottom, and consider only the lattice of nonempty closed sets, we obtain an equivalence between the category ${\bf CONT}\sb\perp$ of continuous dcpo's with bottom and Scott continuous maps as morphisms and the category ${\bf CDL}\sbsp{1}{\prime}$ of completely distributive lattices with maps preserving all nonempty suprema and supprimes as morphisms. If the requirement of preserving supprimes on the morphisms ${\bf CDL}\sbsp{1}{\prime}$ is dropped, we obtain a larger category CDL, and the equivalence above gives an adjunction which generalizes the Hoare power domain The Smyth power domain is another important power domain construction. Given an algebraic dcpo, its Smyth power domain is characterized as the algebraic semilattice of nonempty Scott compact and saturated subsets of the underlying dcpo. We generalize this to nonalgebraic dcpo's and characterize their semilattices of nonempty Scott compact and saturated subsets. First, an equivalence is established between the category QCONT of quasicontinuous dcpo's with Scott continuous maps as morphisms and the category CSL' of the continuous semilattices characterized by the property that the waybelow relation is multiplicative and prime separating, with Scott continuous semilattice primepreserving maps as morphisms. Then several of its subdualities are studied. Finally the requirement that the morphisms preserve primes is dropped, then an adjunction is obtained which generalizes the Smyth power domain By combining the two equivalences thus obtained, we obtain an equivalence between ${\bf CDL}\sbsp{1}{\prime}$ and ${\bf CSL}\sbsp{\rm c}{\prime},$ which is a subcategory of CSL$\sp\prime$ equivalent to ${\bf CONT}\sb\perp.$ This reveals a very interesting relationship between two of the very important power domains 
Language 
eng 
Advisor(s) 
Mislove, Michael 
Degree Date 
1993 
Degree 
Ph.D 
Publisher 
Tulane University 
Publication Date 
1993 
Source 
Source: 67 p., Dissertation Abstracts International, Volume: 5404, Section: B, page: 2009 
Identifier 
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Rights 
Copyright is in accordance with U.S. Copyright law 
Contact Information 
acase@tulane.edu 
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