Weakly meet $s_Z$-continuity and ${\delta }_Z$-continuity

Abstract

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 $\mathbf{POSE}{\mathbf{T}}_{\delta }$ and characterize its $Eilenberg$-$Moore$ algebras concretely.

11 pages