In this note we introduce the poset of $m$-multichains of a given poset $\mathcal{ P}$. Its elements are the multichains of $\mathcal{ P}$ consisting of $m$ elements, and its partial order is the componentwise partial order of $\mathcal{ P}$. We show that this construction preserves a number of poset-theoretic and poset-topological properties of $\mathcal{ P}$. Moreover, we describe the structure of the poset of $m$-multichains of a finite distributive lattice, and provide a link to R.~Stanley's theory of $\mathcal{ P}$-partitions.
Categories: math.CO