Semi unbounded order convergent in ordered vector spaces

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.