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

TL;DR

This paper explores properties of Scott closures in posets, establishing conditions under which weak one-step closure and one-step closure occur, and providing counterexamples and characterizations related to domain theory.

Contribution

It introduces the concepts of weak one-step closure and one-step closure, proves their relationship, and constructs a non-continuous poset with one-step closure answering an open problem.

Findings

Every quasicontinuous domain has weak one-step closure

A non-continuous poset with one-step closure exists

A poset has one-step closure iff it is meet continuous and has weak one-step closure

Abstract

This paper studies the weak one-step closure and one-step closure properties concerning the structure of Scott closures. We deduce that every quasicontinuous domain has weak one-step closure and show that a quasicontinuous poset need not have weak one-step closure. We also constructed a non-continuous poset with one-step closure, which gives a negative answer to an open problem posed by Zou et al.. Finally, we investigate the relationship between weak one-step closure property and one-step closure property and prove that a poset has one-step closure if and only if it is meet continuous and has weak one-step closure.