  Let $\mathcal{P}$ be a poset on $[n]$, $\mathcal{I}(\mathcal{P})$ the set of
order ideals of $\mathcal{P}$ and $E$ an equivalence relation on
$\mathcal{I}(\mathcal{P})$. The concepts of the dual relation $E^*$ of an
equivalence relation $E$, the $E$-weight (resp. $E^*$-weight) distribution of a
linear poset code (resp. its dual poset code) and a MacWilliams-type
equivalence relation are introduced. We give a characterization for a
MacWilliams-type equivalence relation in terms of MacWilliams-type identities
for a linear poset code. Three kinds of equivalence relations on
$\mathcal{I}(\mathcal{P})$ which are of MacWilliams-type are found, i.e., $(i)$
we show that every equivalence relation defined by the automorphism of
$\mathcal{P}$ is a MacWilliams-type; $(ii)$ we provide a new characterization
for poset structures when the equivalence relation defined by the same
cardinality on $\mathcal{I}(\mathcal{P})$ becomes a MacWilliams-type; $(iii)$
we also give necessary and sufficient conditions for poset structures in which
the equivalence relation defined by the order-isomorphism on
$\mathcal{I}(\mathcal{P})$ is a MacWilliams-type.
