Given any subset selection Z$\mathcal {Z}$ for posets, we study two weakenings of the known concept of Z$\mathcal {Z}$-predistributivity, namely, Z$\mathcal {Z}$-quasidistributivity and Z$\mathcal {Z}$-meet-distributivity. The former generalizes quasicontinuity, and the latter meet-continuity of complete lattices. We show for global completions Z$\mathcal {Z}$ that the Z$\mathcal {Z}$-quasidistributive and Z$\mathcal {Z}$-meet-distributive posets are the Z$\mathcal {Z}$-predistributive ones. For the Z$\mathcal {Z}$-Δ-ideal completion ZΔP={Y⊆P:ΔZY=Y}$\mathcal {Z}^{\Delta } P = \{ Y\subseteq P: {\Delta }^{\mathcal {Z}}Y = Y\}$, P$\mathcal {P}$-quasidistributivity is Z$\mathcal {Z}$-quasidistributivity plus ZΔ$\mathcal {Z}^{\Delta }$-quasidistributivity, provided ΔZ${\Delta }^{\mathcal {Z}}$ is idempotent. For Z$\mathcal {Z}$-continuous normal completions e : P → N, we show that Z$\mathcal {Z}$-quasidistributivity of P implies that of N, and the converse holds as well if e is Z$\mathcal {Z}$-initial. This supplements the corresponding results, due to Erné, on the completion-invariance of Z$\mathcal {Z}$-predistributivity and Z$\mathcal {Z}$-meet-distributivity. If Z$\mathcal {Z}$ is a subset system and the Z$\mathcal {Z}$-below relation on the subsets of a poset P has the interpolation property then P is Z$\mathcal {Z}$-quasidistributive and may be embedded in a cube by a map that is ZΔ$\mathcal {Z}^{\Delta }$-continuous and continuous for the lower topologies.

中文翻译：

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Z $\mathcal {Z}$ -quasidistributive 和 Z $\mathcal {Z}$ -meet-distributive Posets

给定偏序集的任何子集选择 Z$\mathcal {Z}$，我们研究了已知概念 Z$\mathcal {Z}$-predistributivity 的两个弱点，即 Z$\mathcal {Z}$-quasidistributivity 和 Z$ \mathcal {Z}$-meet-distributivity。前者概括了拟连续性，后者满足了完全格的连续性。对于全局补全 Z$\mathcal {Z}$，我们证明 Z$\mathcal {Z}$-quasidistributive 和 Z$\mathcal {Z}$-meet-distributive 位组是 Z$\mathcal {Z}$-预分配的。对于 Z$\mathcal {Z}$-Δ-理想补全 ZΔP={Y⊆P:ΔZY=Y}$\mathcal {Z}^{\Delta } P = \{ Y\subseteq P: {\Delta } ^{\mathcal {Z}}Y = Y\}$, P$\mathcal {P}$-quasidistributivity 是 Z$\mathcal {Z}$-quasidistributivity 加上 ZΔ$\mathcal {Z}^{\Delta }$ - 准分配性，前提是 ΔZ${\Delta }^{\mathcal {Z}}$ 是幂等的。对于 Z$\mathcal {Z}$-连续正常完成 e : P → N，我们证明 Z$\mathcal {Z}$-P 的拟分配性暗示了 N 的拟分配性，如果 e 是 Z$，则反之亦然\mathcal {Z}$-初始。这补充了 Erné 对 Z$\mathcal {Z}$-predistributivity 和 Z$\mathcal {Z}$-meet-distributivity 的完成不变性的相应结果。如果 Z$\mathcal {Z}$ 是一个子集系统，并且偏序集 P 的子集上的 Z$\mathcal {Z}$-below 关系具有插值性质，那么 P 是 Z$\mathcal {Z}$-quasidistributive并且可以通过 ZΔ$\mathcal {Z}^{\Delta }$-连续和对于较低拓扑连续的映射嵌入立方体中。这补充了 Erné 对 Z$\mathcal {Z}$-predistributivity 和 Z$\mathcal {Z}$-meet-distributivity 的完成不变性的相应结果。如果 Z$\mathcal {Z}$ 是一个子集系统，并且偏序集 P 的子集上的 Z$\mathcal {Z}$-below 关系具有插值性质，那么 P 是 Z$\mathcal {Z}$-quasidistributive并且可以通过 ZΔ$\mathcal {Z}^{\Delta }$-连续和对于较低拓扑连续的映射嵌入立方体中。这补充了 Erné 对 Z$\mathcal {Z}$-predistributivity 和 Z$\mathcal {Z}$-meet-distributivity 的完成不变性的相应结果。如果 Z$\mathcal {Z}$ 是一个子集系统，并且偏序集 P 的子集上的 Z$\mathcal {Z}$-below 关系具有插值性质，那么 P 是 Z$\mathcal {Z}$-quasidistributive并且可以通过 ZΔ$\mathcal {Z}^{\Delta }$-连续和对于较低拓扑连续的映射嵌入立方体中。