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 $x_\alpha\xrightarrow {\{W, T\}}x$), 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 $\pm (Tx_\alpha -Tx) \leq y_\beta$ whenever $\alpha \geq \alpha_0$. 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 $x_\alpha\xrightarrow {\{{W_1}, T_1\}}x$ implies $Sx_\alpha\xrightarrow {\{W_2, T_2\}}Sx$ whenever $\{x_\alpha\}\subseteq V_1$. We study some properties of this new classification of operators.