A generalization of order continuous operators

Abstract

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.