A generalization of order convergence

Let $E$ be a sublattice of a vector lattice $F$. $\left( x_\alpha \right)\subseteq E$ is said to be $F$-order convergent to a vector $x$ (in symbols $x_\alpha \xrightarrow{Fo} x$), whenever there exists another net $\left(y_\alpha\right)$ in $F$ with the some index set satisfying $y_\alpha\downarrow 0$ in $F$ and $| x_\alpha - x | \leq y_\alpha$ for all indexes $\alpha$. If $F=E^{\sim\sim}$, this convergence is called $b$-order convergence and we write $x_\alpha \xrightarrow{bo} x$. In this manuscript, first we study some properties of $Fo$-convergence nets and we extend some results to the general case. In the second part, we introduce $b$-order continuous operators and we invistegate some properties of this new concept. An operator $T$ between two vector lattices $E$ and $F$ is said to be $b$-order continuous, if $x_\alpha \xrightarrow{bo} 0$ in $E$ implies $Tx_\alpha \xrightarrow{bo} 0$ in $F$.