A generalization of order continuous operators

Abstract

Let $E$ be a sublattice of a vector lattice $F$. A net $\{ x_α\}_{α\in \mathcal{A}}\subseteq E$ is said to be $ F $-order convergent to a vector $ x \in E$ (in symbols $ x_α\xrightarrow{Fo} x $ in $E$), whenever there exists a net $ \{y_β\}_{β\in \mathcal{B}} $ in $F $ satisfying $ y_β\downarrow 0 $ in $F$ and for each $β$, there exists $α_0$ such that $ \vert x_α- x \vert \leq y_β$ whenever $ α\geq α_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 $x_α\xrightarrow{Fo} 0$ in $E$ implies $Tx_α\xrightarrow{Ho} 0$ in $G$. We will study some properties of this new classification of operators and its relationships with order continuous operators.