Transporting continuity properties from a poset to its subposets

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.

42 pages. For the final publication, see https://www.sciencedirect.com/science/article/abs/pii/S0304397522001013

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• How regular can maxitive measures be?
• Domain theory and mirror properties in inverse semigroups