Transporting continuity properties from a poset to its subposets

TL;DR

This paper investigates conditions under which continuity properties of a poset can be transferred to its subposets, introducing the concept of way-below preserving subposets and their significance in the structure of posets.

Contribution

It identifies key conditions for transferring continuity properties to subposets and introduces way-below preserving subposets within the Z theory framework.

Findings

Conditions for transferring continuity properties are established.

Existence of a largest continuous way-below preserving subposet in certain posets.

Most results are formulated within the general Z theory setting.

Abstract

We identify two key conditions that a subset $A$ of a poset $P$ may satisfy to guarantee the transfer of continuity properties from $P$ to $A$. We then highlight practical cases where these key conditions are fulfilled. Along the way we are led to consider subsets of a given poset $P$ whose way-below relation is the restriction of the way-below relation of $P$, which we call way-below preserving subposets. As an application, we show that every conditionally complete poset with the interpolation property contains a largest continuous way-below preserving subposet. Most of our results are expressed in the general setting of Z theory, where Z is a subset system.