Semi-order continuous operators on vector spaces

Abstract

In this manuscript, we will study both $\tilde{o}$-convergence in (partially) ordered vector spaces and a kind of convergence in a vector space $V$. A vector space $V$ is called semi-order vector space (in short semi-order space), if there exist an ordered vector space $W$ and an operator $T$ from $V$ into $W$. In this way, we say that $V$ is semi-order space with respect to $\{W,T\}$. A net $\{x_{\alpha }\}\subseteq V$ is said to be $\{W,T\}$-order convergent to a vector $x\in V$ (in short we write $x_{\alpha }\stackrel{\{W,T\}}{\to }x$), whenever there exists a net $\{y_{\beta }\}$ in $W$ satisfying $y_{\beta }\downarrow 0$ in $W$ and for each $\beta$, there exists ${\alpha }_0$ such that $\pm (T{x}_{\alpha }-Tx)\le y_{\beta }$ whenever $\alpha \ge {\alpha }_0$. In this manuscript, we study and investigate some properties of $\{W,T\}$-convergent nets and its relationships with other order convergence in partially ordered vector spaces. Assume that $V_1$ and $V_2$ are semi-order spaces with respect to $\{W_1,T_1\}$ and $\{W_2,T_2\}$, respectively. An operator $S$ from $V_1$ into $V_2$ is called semi-order continuous, if $x_{\alpha }\stackrel{\{W_1,T_1\}}{\to }x$ implies $S{x}_{\alpha }\stackrel{\{W_2,T_2\}}{\to }Sx$ whenever $\{x_{\alpha }\}\subseteq V_1$. We study some properties of this new classification of operators.