### A GENERALIZATION OF ORDER CONVERGENCE IN THE VECTOR LATTICES

DOI Number
https://doi.org/10.22190/FUMI210417036H
First page
521
Last page
528

#### 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 $F$.

#### Keywords

order convergence, vector lattice, continuous operator

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#### References

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DOI: https://doi.org/10.22190/FUMI210417036H

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