Semi unbounded order convergent in ordered vector spaces

TL;DR

This paper introduces semi unbounded order convergence in ordered vector spaces, explores its properties, and compares it with existing order convergence notions, providing a new perspective in the theory of ordered vector spaces.

Contribution

It defines semi unbounded order convergence, analyzes its properties, and establishes its equivalence with unbounded order convergence in vector lattices.

Findings

Semi unbounded order convergence is characterized by a net and a decreasing net tending to zero.

In vector lattices, semi unbounded order convergence coincides with unbounded order convergence.

The paper elucidates relationships between semi unbounded order convergence and other known order convergences.

Abstract

Let $X$ be an ordered vector space. The net $\{x_\alpha\}\subseteq X$ is semi unbounded order convergent to $x$ (in symbol $x_\alpha\xrightarrow{suo}x$), if there is a net $\{y_\beta\}$, possibly over a different index set, such that $y_\beta \downarrow 0$ and for every $\beta$ there exists $\alpha_0$ such that $\{\{\pm(x_\alpha - x)\}^u,y\}^l\subseteq \{y_\beta\}^l$, whenever $\alpha \geq \alpha_0$ and for all $0\leq y \in X$. In vector lattice $E$, semi unbounded order convergence is equivalent with unbounded order convergence. We study some properties of this convergence and some of its relationships with others known order convergence.