  We define bidual bounded $uo$-convergence in vector lattices and investigate
relations between this convergence and $b$-property. We prove that for a
regular Riesz dual system $\langle X,X^{\sim}\rangle$, $X$ has $b$-property if
and only if the order convergence in $X$ agrees with the order convergence in
$X^{\sim\sim}$.
