Quasicontinuity of Posets via Scott Topology and Sobrification

Research paper by Xuxin Mao, Luoshan Xu

Indexed on: 20 Jan '07Published on: 20 Jan '07Published in: Order


In this paper, posets which may not be dcpos are considered. In terms of the Scott topology on posets, the new concept of quasicontinuous posets is introduced. Some properties and characterizations of quasicontinuous posets are examined. The main results are: (1) a poset is quasicontinuous iff the lattice of all Scott open sets is a hypercontinuous lattice; (2) the directed completions of quasicontinuous posets are quasicontinuous domains; (3) A poset is continuous iff it is quasicontinuous and meet continuous, generalizing the relevant result for dcpos.