The m-cover posets and their applications

In this article we introduce the m  -cover poset of an arbitrary bounded poset PP, which is a certain subposet of the m  -fold direct product of PP with itself. Its ground set consists of multichains of PP that contain at most three different elements, one of which has to be the least element of PP, and the other two elements have to form a cover relation in PP. 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>0m>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 TnTn, and we show that the smallest lattice that contains the m  -cover poset of TnTn is isomorphic to the m  -Tamari lattice Tn(m) introduced by Bergeron and Préville-Ratelle. We conclude this article with a conjectural desription of an explicit realization of Tn(m) in terms of m-tuples of Dyck paths.