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

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#### 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 $F$.

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Y. A. Abramovich and C. D. Aliprantis: Locally Solid vector lattices with Application to Economics: Mathematical Surveys, 105, American Mathematical Society, Providence, RI, 2003.

C. D. Aliprantis and O. Burkinshaw: Positive operators, 119, Springer Science & Business Media, 2006.

S¸. Alpay, B. Altin and C. Tonyali: On property (b) of vector lattices: Positivity. 7 (2003), 135–139.

S¸. Alpay; E. Yu. Emel’yanov and Z. Ercan: A characterization of an order ideal in Riesz spaces. Proc. Amer. Math. Soc. 132 (2004), 3627-3628.

S¸. Alpay and S. Gorokhova: b-property of sublattices in vector lattices: Turkish J. Math. 45 (2021), 1555-1563.

S¸. Alpay and Z. Ercan: Characterizations of Riesz spaces with b-property: Positivity. 13 (2009), 21–30.

B. Aqzzouz, A. Elbour and J. Hmichane: The duality problem for the class of b-weakly compact operators: Positivity. 13 (2009), 683–692.

M. Kandic, H. Li and V. G. Troitsky: Unbounded norm topology beyond normed lattices: Positivity. 22 (2018), 745-760.

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

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