# Partial dcpo’s and some applications

## Zhao Dongsheng

Address: Mathematics and Mathematics Education, National Institute of Education Singapore, Nanyang Technological University, 1 Nanyang Walk, Singapore 637616

E-mail: dongsheng.zhao@nie.edu.sg

Abstract: 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.

AMSclassification: primary 06B35; secondary 06F30, 06B23, 54C05, 54C30, 54A05.

Keywords: directed complete poset, Scott topology, dcpo-completion, partial dcpo, C-space, lattice of continuous functions, lower semicontinuous functions, injective hull.

DOI: 10.5817/AM2012-4-243