b-Property of sublattices in vector lattices

We study $b$-property of a sublattice (or an order ideal) $F$ of a vector lattice $E$. In particular, $b$-property of $E$ in $E^δ$, the Dedekind completion of $E$, $b$-property of $E$ in $E^u$, the universal completion of $E$, and $b$-property of $E$ in $\hat{E}(\hatτ)$, the completion of $E$.

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