Domains via approximation operators

Abstract

In this paper, we tailor-make new approximation operators inspired by rough set theory and specially suited for domain theory. Our approximation operators offer a fresh perspective to existing concepts and results in domain theory, but also reveal ways to establishing novel domain-theoretic results. For instance, (1) the well-known interpolation property of the way-below relation on a continuous poset is equivalent to the idempotence of a certain set-operator; (2) the continuity of a poset can be characterized by the coincidence of the Scott closure operator and the upper approximation operator induced by the way below relation; (3) meet-continuity can be established from a certain property of the topological closure operator. Additionally, we show how, to each approximating relation, an associated order-compatible topology can be defined in such a way that for the case of a continuous poset the topology associated to the way-below relation is exactly the Scott topology. A preliminary investigation is carried out on this new topology.

One-step closure, weak one-step closure and meet continuity

“…A poset P is said to have the weak one-step closure if for any A ⊆ P , it holds that cl(A) = A ′′ , whereA ′′ = {x ∈ P | ∃D ⊆ ↑ ↓A, x ≤ sup D} Remark 3.2 In [7], Zhao introduced the Definition 3.1 for an arbitrary set system, and called it one-step closure. To be consistent with the paper[6], here we call this property weak one-step closure. Every quasicontinuous dcpo has weak one-step closure.…”

“…In Section 3, we prove that every quasicontinuous domain has weak one-step closure and show, by a counterexample, that a quasicontinuous poset may not have weak one-step closure. In Section 4, we give a negative answer to the problem posed by Zou et al in [6]. In Section 5, we prove that a poset has one-step closure if and only if it is meet continuous and has weak one-step closure.…”

“…For any quasicontinuous domain P , the family { ↑ ↑F : F ⊆ P is finite} is a base of the Scott topology on P ( [2]). 3 Weak one-step closure By [6], every continuous poset has one-step closure. However, a quasicontinuous poset may not have onestep closure.…”

“…In [6], Zou, Li and Ho showed that every continuous poset has one-step closure. They asked whether L is continuous if it has one-step closure.…”

“…In [7], Zhao introduced the weak one-step closure property in order to obtain some characterizations of Z-continuous posets. In [6], Zou et al proposed the one-step closure property and proved that every continuous poset has one-step closure. They asked whether all posets with one-step closure are continuous.…”