A GENERALIZATION OF ORDER CONVERGENCE IN THE VECTOR LATTICES

Abstract

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 $ \vert x_\alpha - x \vert \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 same 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 $F$.