Quasicontinuity of Posets via Scott Topology and Sobrification

Abstract

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.

D-completions and thed-topology

“…Remark 7.7. As remarked in the introduction, the preceding definition is, in general, distinct from the directed completion of a poset introduced by Xu [24], which also appears in [25] and Mao and Xu [20]. There the directed completion of a poset is taken to be the space of irreducible Scott-closed subsets with the inclusion order.…”

“…Several of our results in the order theoretic setting, which we derive as corollaries of our topological theory, were previously obtained by them. A related, but distinctly different, approach was taken by Xu [24,25] and Mao and Xu [20], who defined the directed completion of a poset to be the space of irreducible Scottclosed subsets with the inclusion order. The latter is one manifestation of the sobrification (with the order of specialization) of the original poset equipped with the Scott topology, and is in general distinctly larger than the D-completion.…”

A Poset with Spectral Scott Topology is a Quasialgebraic Domain

“…According to [11], a poset X is called a quasicontinuous poset (resp. quasialgebraic poset) if for all x ∈ U ∈ σ (X), there is a nonempty finite F ⊆ X such that…”

s2-Quasicontinuous posets

“…An important direction in the study of continuous domains is to extend the theory of continuous domains to that of posets as much as possible [12,[14][15][16]20,22]. In [3], Erné introduced the concept of s 2 -continuous posets by making use of the cut operator instead of joins.…”