## $$\mathcal{Z}$$-quasidistributive and $$\mathcal{Z}$$-meet-distributive posets.(English)Zbl 1483.06006

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.

### MSC:

 06A06 Partial orders, general 06B23 Complete lattices, completions 06D75 Other generalizations of distributive lattices
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### References:

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