  Based on the concept of weakly meet $s_{Z}$-continuouity put forward by Xu
and Luo in \cite{qzm}, we further prove that if the subset system $Z$ satisfies
certain conditions, a poset is $s_{Z}$-continuous if and only if it is weakly
meet $s_{Z}$-continuous and $s_{Z}$-quasicontinuous, which improves a related
result given by Ruan and Xu in \cite{sz}. Meanwhile, we provide a
characterization for the poset to be weakly meet $s_{Z}$-continuous, that is, a
poset with a lower hereditary $Z$-Scott topology is weakly meet
$s_{Z}$-continuous if and only if it is locally weakly meet $s_{Z}$-continuous.
In addition, we introduce a monad on the new category $\mathbf{POSET_{\delta}}$
and characterize its $Eilenberg$-$Moore$ algebras concretely.
