Recent zbMATH articles in MSC 06D75 https://www.zbmath.org/atom/cc/06D75 2022-05-16T20:40:13.078697Z Werkzeug $$\mathcal{Z}$$-quasidistributive and $$\mathcal{Z}$$-meet-distributive posets https://www.zbmath.org/1483.06006 2022-05-16T20:40:13.078697Z "Zhang, Wenfeng" https://www.zbmath.org/authors/?q=ai:zhang.wenfeng.1|zhang.wenfeng "Xu, Xiaoquan" https://www.zbmath.org/authors/?q=ai:xu.xiaoquan.1|xu.xiaoquan In domain theory, one fundamental result states that a poset is continuous if and only if it is quasicontinuous and meet-continuous. In this paper, the authors study two kinds of distributivity: $$Z$$-quasidistributivity and $$Z$$-meet-distributivity, which are the generalizations of quasicontinuity and meet-continuity. Here, $$Z$$ is a subset system, it becomes meaningful when $$Z$$ is replaced by adjectives such as directed'', chain'', finite'', etc. Analogous to the above fundamental result, the authors prove that, under some conditions, a poset is $$Z$$-predistributive iff it is $$Z$$-quasidistributive and $$Z$$-meet-distributive. In order theory, the Dedekind-MacNeille completion is the most well-known completion, which embeds a poset into a complete lattice. The order-theoretical properties which are invariant under the Dedekind-MacNeille completion are called completion-invariant. In this paper, one main result states that $$Z$$-quasidistributivity is a completion-invariant property whenever $$Z$$ is completion-stable. The way-below relation is a fundamental concept in domain theory. Replacing directed sets by $$Z$$-sets, one has the concept of $$Z$$-below. The last main result of this paper: if the $$Z$$-below relation on the subsets of a poset $$P$$ has the interpolation property, then $$P$$ is embeddable in a cube. Reviewer: Zhongxi Zhang (Yantai)