LAUSR

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.