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\(\mathcal {Z}\)-quasidistributive and \(\mathcal {Z}\)-meet-distributive Posets

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Abstract

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 \(\mathcal {Z}\)-Δ-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 → 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 Erné, 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.

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Acknowledgements

We are indebted to Marcel Erné, who directed our attention to the useful notions of \(\mathcal {Z}\)-continuity and \(\mathcal {Z}\)-initiality, and gave many valuable hints that have improved considerably the first draft of this paper, in which most of the results were established only for the case \(\mathcal {Z} = \mathcal {P}_{m}\) or \(\mathcal {Z} = \mathcal {P}_{m}^{\ {\Delta }}\).

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Authors and Affiliations

  1. School of Mathematics and Computer Science, Jiangxi Science & Technology Normal University, Nanchang, 330038, China

    Wenfeng Zhang

  2. School of Mathematics and Statistics, Minnan Normal University, Zhangzhou, 363000, China

    Xiaoquan Xu

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  1. Wenfeng Zhang
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  2. Xiaoquan Xu
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Correspondence to Wenfeng Zhang.

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Supported by the National Natural Science Foundation of China (Nos. 11701238, 11661057), the Natural Science Foundation of Jiangxi Province (Nos. 20161BAB211017, 20161BAB2061004) and the Young Talent Support Plan of Jiangxi Science and Technology Normal University (No. 2016QNBJRC008).

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Zhang, W., Xu, X. \(\mathcal {Z}\)-quasidistributive and \(\mathcal {Z}\)-meet-distributive Posets. Order 37, 103–113 (2020). https://doi.org/10.1007/s11083-019-09495-2

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