A generalization of order continuous operators

Let $E$ be a sublattice of a vector lattice $F$. A net $\{ x_\alpha \}_{\alpha \in \mathcal{A}}\subseteq E$ is said to be $F$-order convergent to a vector $x \in E$ (in symbols $x_\alpha \xrightarrow{Fo} x$ in $E$), whenever there exists a net $\{y_\beta\}_{\beta \in \mathcal{B}}$ in $F$ satisfying $y_\beta\downarrow 0$ in $F$ and for each $\beta$, there exists $\alpha_0$ such that $\vert x_\alpha - x \vert \leq y_\beta$ whenever $\alpha \geq \alpha_0$. In this manuscript, first we study some properties of $F$-order convergence nets and we extend some results to the general cases. Let $E$ and $G$ be sublattices of vector lattices $F$ and $H$ respectively. We introduce $FH$-order continuous operators, that is, an operator $T$ between two vector lattices $E$ and $G$ is said to be $FH$-order continuous, if $x_\alpha \xrightarrow{Fo} 0$ in $E$ implies $Tx_\alpha \xrightarrow{Ho} 0$ in $G$. We will study some properties of this new classification of operators and its relationships with order continuous operators.