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The $m$-Cover Posets and Their Applications

In this article we introduce the $m$-cover poset of an arbitrary bounded poset $\mathcal{P}$, which is a certain subposet of the $m$-fold direct product of $\mathcal{P}$ with itself. Its ground set consists of multichains of $\mathcal{P}$ that contain at most three different elements, one of which has to be the least element of $\mathcal{P}$, and the other two elements have to form a cover relation in $\mathcal{P}$. We study the $m$-cover poset from a structural and topological point of view. In particular, we characterize the posets whose $m$-cover poset is a lattice for all $m>0$, and we characterize the special cases, where these lattices are EL-shellable, left-modular, or trim. Subsequently, we investigate the $m$-cover poset of the Tamari lattice $\mathcal{T}_{n}$, and we show that the smallest lattice that contains the $m$-cover poset of $\mathcal{T}_{n}$ is isomorphic to the $m$-Tamari lattice $\mathcal{T}_{n}^{(m)}$ introduced by Bergeron and Préville-Ratelle. We conclude this article with a conjectural desription of an explicit realization of $\mathcal{T}_{n}^{(m)}$ in terms of $m$-tuples of Dyck paths.