Recent zbMATH articles in MSC 06D75https://www.zbmath.org/atom/cc/06D752022-05-16T20:40:13.078697ZWerkzeug\( \mathcal{Z} \)-quasidistributive and \(\mathcal{Z} \)-meet-distributive posetshttps://www.zbmath.org/1483.060062022-05-16T20:40:13.078697Z"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.xiaoquanIn 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)