Various types of completeness in topologized semilattices

TL;DR

This paper explores different notions of completeness in topologized semilattices, introducing new concepts, analyzing their properties, and providing examples to distinguish these classes, thereby extending existing results in the field.

Contribution

It introduces various new concepts of completeness in topologized semilattices using generalized closure operators and studies their fundamental properties and distinctions.

Findings

Different classes of completeness do not coincide.

New theorems generalize results on complete semilattices with topology.

Examples illustrate the distinctions among the classes.

Abstract

A topologized semilattice $X$ is called complete if each non-empty chain $C\subset X$ has $\inf C$ and $\sup C$ that belong to the closure $C$ of the chain $C$ in $X$. In this paper, we introduce various concepts of completeness of topologized semilattices in the context of operators that generalize the closure operator, and study their basic properties. In addition, examples of specific topologized semilattices are given, showing that these classes do not coincide with each other. Also in this paper, we prove theorems that allow us to generalize the available results on complete semilattices endowed with topology.