an:07204027
Zbl 7204027
Zhang, Wenfeng; Xu, Xiaoquan
\( \mathcal{Z} \)-quasidistributive and \(\mathcal{Z} \)-meet-distributive posets
EN
Order 37, No. 1, 103-113 (2020)
0167-8094 1572-9273
2020
j
06-XX
completion-invariant; \( \mathcal{Z} \)-initial; \( \mathcal{Z} \)-predistributive; \( \mathcal{Z} \)-quasidistributive; \( \mathcal{Z} \)-meet-distributive; normal completion
Summary: Given any subset selection \(\mathcal{Z}\) for posets, we study two weakenings of the known concept of \(\mathcal{Z} \)-predistributivity, namely, \( \mathcal{Z} \)-quasidistributivity and \(\mathcal{Z} \)-meet-distributivity. The former generalizes quasicontinuity, and the latter meet-continuity of complete lattices. We show for global completions \(\mathcal{Z}\) that the \(\mathcal{Z} \)-quasidistributive and \(\mathcal{Z} \)-meet-distributive posets are the \(\mathcal{Z} \)-predistributive ones. For the \(\mathcal{Z}\)-\(\Delta \)-ideal completion \(\mathcal{Z}^{\Delta } P = \{ Y\subseteq P: {\Delta }^{\mathcal{Z}}Y = Y\}, \mathcal{P} \)-quasidistributivity is \(\mathcal{Z} \)-quasidistributivity plus \(\mathcal{Z}^{\Delta } \)-quasidistributivity, provided \({\Delta }^{\mathcal{Z}}\) is idempotent. For \(\mathcal{Z} \)-continuous normal completions \(e : P \rightarrow N\), we show that \(\mathcal{Z} \)-quasidistributivity of \(P\) implies that of \(N\), and the converse holds as well if \(e\) is \(\mathcal{Z} \)-initial. This supplements the corresponding results, due to ErnĂ©, on the completion-invariance of \(\mathcal{Z} \)-predistributivity and \(\mathcal{Z} \)-meet-distributivity. If \(\mathcal{Z}\) is a subset system and the \(\mathcal{Z} \)-below relation on the subsets of a poset \(P\) has the interpolation property then \(P\) is \(\mathcal{Z} \)-quasidistributive and may be embedded in a cube by a map that is \(\mathcal{Z}^{\Delta } \)-continuous and continuous for the lower topologies.