# A generalization of order convergence

## TL;DR

This paper introduces and studies a new form of order convergence called $F$-order convergence, explores its properties, and defines $b$-order continuous operators, analyzing their characteristics in vector lattices.

## Contribution

It generalizes order convergence by defining $F$-order convergence and introduces $b$-order continuous operators, expanding the theoretical framework of vector lattice analysis.

## Key findings

- Properties of $Fo$-convergence nets are established.
- Extension of results to the general case of $Fo$-convergence.
- Introduction and analysis of $b$-order continuous operators.

## 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 $ | 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 $F$.