  We generalize the notion of $M$-ideals in order smooth $\infty$-normed spaces
to "smooth $p$-order ideals" in order smooth $p$-normed spaces. We show that if
$V$ is an order smooth $p$-normed space and $W$ is a closed subspace of $V$,
then $W$ is a smooth $p$-order ideal in $V$ if and only if $W^{\perp}$ is a
smooth $p'$-order ideal in order smooth $p'$-normed space if and only if
$W^{\perp\perp}$ is a smooth $p$-order ideal in order smooth $p$-normed space
$V^{**}$. We prove that every $L$-summand in order smooth $1$-normed space is a
smooth $1$-order ideal. We find a condition under which every $M$-ideal in
order smooth $\infty$-normed space is a smooth $\infty$-order ideal. We show
that every $M$-ideal in order smooth $\infty$-normed space is smooth
$\infty$-order ideal.
