Weakly meet $s_{Z}$-continuity and $\delta_{Z}$-continuity

TL;DR

This paper explores the relationships between different forms of continuity in posets, providing new characterizations and conditions under which these forms are equivalent, and introduces a monad on a category of posets.

Contribution

It establishes equivalences between $s_{Z}$-continuity, weakly meet $s_{Z}$-continuity, and $s_{Z}$-quasicontinuity under certain conditions, and characterizes weakly meet $s_{Z}$-continuity via local properties, also introducing a monad on $ extbf{POSET}_{oldsymbol{ extdelta}}$.

Findings

Poset $s_{Z}$-continuity is equivalent to weakly meet $s_{Z}$-continuity and $s_{Z}$-quasicontinuity under certain conditions.

A poset with a lower hereditary $Z$-Scott topology is weakly meet $s_{Z}$-continuous iff it is locally weakly meet $s_{Z}$-continuous.

A monad on $ extbf{POSET}_{oldsymbol{ extdelta}}$ is introduced and its Eilenberg-Moore algebras are characterized.

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 $\mathbf{POSET_{\delta}}$ and characterize its $Eilenberg$-$Moore$ algebras concretely.