Semi-order continuous operators on vector spaces

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 $\{x_α\}\subseteq V$ is said to be ${\{W,T\}}$-order convergent to a vector $x\in V$ (in short we write $x_α\xrightarrow {\{W, T\}}x$), whenever there exists a net $\{y_β\}$ in $W$ satisfying $y_β\downarrow 0$ in $W$ and for each $β$, there exists $α_0$ such that $\pm (Tx_α-Tx) \leq y_β$ whenever $α\geq α_0$. 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 $x_α\xrightarrow {\{{W_1}, T_1\}}x$ implies $Sx_α\xrightarrow {\{W_2, T_2\}}Sx$ whenever $\{x_α\}\subseteq V_1$. We study some properties of this new classification of operators.

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