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 $E^{\delta }$, the Dedekind completion of $E$, $b$-property of $E$ in $E^u$, the universal completion of $E$, and $b$-property of $E$ in $\widehat{E}(\widehat{\tau })$, the completion of $E$.