Structural Properties of the Cambrian Semilattices -- Consequences of Semidistributivity

The $\gamma$-Cambrian semilattices $\mathcal{C}_{\gamma}$ defined by Reading and Speyer are a family of meet-semilattices associated with a Coxeter group $W$ and a Coxeter element $\gamma\in W$, and they are lattices if and only if $W$ is finite. In the case where $W$ is the symmetric group $\mathfrak{S}_{n}$ and $\gamma$ is the long cycle $(1\;2\;\ldots\;n)$ the corresponding $\gamma$-Cambrian lattice is isomorphic to the well-known Tamari lattice $\mathcal{T}_{n}$. Recently, Kallipoliti and the author have investigated $\mathcal{C}_{\gamma}$ from a topological viewpoint, and showed that many properties of the Tamari lattices can be generalized nicely. In the present article this investigation is continued on a structural level using the observation of Reading and Speyer that $\mathcal{C}_{\gamma}$ is semidistributive. First we prove that every closed interval of $\mathcal{C}_{\gamma}$ is a bounded-homomorphic image of a free lattice (in fact it is a so-called $\mathcal{H\!H}$-lattice). Subsequently we prove that each closed interval of $\mathcal{C}_{\gamma}$ is trim, we determine its breadth, and we characterize the closed intervals that are dismantlable.