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- W2430971334 abstract "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 $gammain W$, 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." @default.
- W2430971334 created "2016-06-24" @default.
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- W2430971334 date "2013-12-16" @default.
- W2430971334 modified "2023-09-27" @default.
- W2430971334 title "Structural Properties of the Cambrian Semilattices -- Consequences of Semidistributivity" @default.
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