Order continuity from a topological perspective

TL;DR

This paper explores order convergence and order continuity in various algebraic structures, introducing an order topology that makes order continuity a topological property under mild conditions, and generalizes a key theorem on order continuous operators.

Contribution

It introduces an order topology framework for partially ordered structures and generalizes the Ogasawara theorem on order continuous operators.

Findings

Order continuity becomes a topological property in certain structures.

A generalized Ogasawara theorem is established.

Order convergence concepts are extended to multiple algebraic settings.

Abstract

We study three types of order convergence and related concepts of order continuous maps in partially ordered sets, partially ordered abelian groups and partially ordered vector spaces, respectively. An order topology is introduced such that in the latter two settings under mild conditions order continuity is a topological property. We present a generalisation of the Ogasawara theorem on the structure of the set of order continuous operators.