  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 $F$.
