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The absolute order is a natural partial order on a Coxeter group W. It can be viewed as an analogue of the weak order on W in which the role of the generating set of simple reflections in W is played by the set of all reflections in W. By use of a notion of constructibility for partially ordered sets, it is proved that the absolute order on the symmetric… (More)

The absolute order on the hyperoctahedral group Bn is investigated. It is shown that every closed interval in this order is shellable, those closed intervals which are lattices are characterized and their zeta polynomials are computed. Moreover, using the notion of strong constructibility, it is proved that the order ideal generated by the Coxeter elements… (More)

For an arbitrary Coxeter group W , Reading and Speyer defined Cambrian semi-lattices C γ as sub-semilattices of the weak order on W induced by so-called γ-sortable elements. In this article, we define an edge-labeling of C γ , and show that this is an EL-labeling for every closed interval of C γ. In addition, we use our labeling to show that every finite… (More)

In this paper we present a bijection between two well known families of Catalan objects: the set of facets of the m-generalized cluster complex ∆ m (A n) and that of dominant regions in the m-Catalan arrangement Cat m (A n), where m ∈ N >0. In particular, the map which we define bijects facets containing the negative simple root −α to dominant regions… (More)

In the first part of this article we present a realization of the m-Tamari lattice T (m) n in terms of m-tuples of Dyck paths of height n, equipped with componentwise rotation order. For that, we define the m-cover poset P m of an arbitrary bounded poset P, and show that the smallest lattice completion of the m-cover poset of the Tamari lattice Tn is… (More)

For an arbitrary Coxeter group W , David Speyer and Nathan Reading defined Cambrian semilattices Cγ as certain sub-semilattices of the weak order on W. In this article, we define an edge-labeling using the realization of Cambrian semilattices in terms of γ-sortable elements, and show that this is an EL-labeling for every closed interval of Cγ. In addition,… (More)

In this paper we study topological properties of the poset of injective words and the lattice of classical non-crossing partitions. Specifically, it is shown that after the removal of the bottom and top elements (if existent) these posets are doubly Cohen-Macaulay. This extends the well-known result that those posets are shellable. Both results rely on a… (More)

The absolute order is a natural partial order on a Coxeter group W. It can be viewed as an analogue of the weak order on W in which the role of the generating set of simple reflections in W is played by the set of all reflections in W. By use of a notion of constructibility for partially ordered sets, it is proved that the absolute order on the symmetric… (More)

We show that locally connected, simply connected homogeneous continua are not separated by arcs. We ask several questions about homogeneous continua which are inspired by analogous questions in geometric group theory.

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