Geodesic
Partial dcpo’s and some applications
Archivum mathematicum, Tome 48 (2012) no. 4, pp. 243-260.

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We introduce partial dcpo’s and show their some applications. A partial dcpo is a poset associated with a designated collection of directed subsets. We prove that (i) the dcpo-completion of every partial dcpo exists; (ii) for certain spaces $X$, the corresponding partial dcpo’s of continuous real valued functions on $X$ are continuous partial dcpos; (iii) if a space $X$ is Hausdorff compact, the lattice of all S-lower semicontinuous functions on $X$ is the dcpo-completion of that of continuous real valued functions on the space; (iv) a topological space has an injective hull iff it is homeomorphic to the pre-Scott space of a continuous partial dcpo whose way-below relation satisfies the interpolation property.
DOI : 10.5817/AM2012-4-243
Classification : 06B23, 06B35, 06F30, 54A05, 54C05, 54C30
Keywords: directed complete poset; Scott topology; dcpo-completion; partial dcpo; C-space; lattice of continuous functions; lower semicontinuous functions; injective hull 
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     title = {Partial dcpo{\textquoteright}s and some applications},
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Dongsheng, Zhao. Partial dcpo’s and some applications. Archivum mathematicum, Tome 48 (2012) no. 4, pp. 243-260. doi : 10.5817/AM2012-4-243. http://geodesic.mathdoc.fr/articles/10.5817/AM2012-4-243/

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