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MacWilliams-type equivalence relations

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.