Lattices from Pointed Building Sets: Generalized Ornamentation Lattices

TL;DR

Pointed building sets yield ornamentation lattices $\mathcal{O}(\mathsf{B})$, unifying Tamari, affine Tamari, topologies, and naturally labeled posets under a single combinatorial construction. The paper proves that $\mathcal{O}(\mathsf{B})$ is a complete lattice with explicit meet and join, and develops inverse-limit frameworks that realize infinite Tamari lattices as limits of finite cases. It also investigates digraphical ornamentations, dualities for directed trees, and invariance under group actions, linking to cyclic and affine Tamari variants, while offering geometric interpretations via root-system-like constructions and discussing polytopal questions. Collectively, these results establish a broad, versatile theory of ornamentation lattices with multiple concrete applications and open directions for future work in combinatorics, geometry, and representation theory.

Abstract

We introduce a novel combinatorial structure called pointed building sets, which can be viewed as families of lattices equipped with compatibility relations. To each pointed building set $\mathsf{B}$, we associate a complete lattice $\mathcal{O}(\mathsf{B})$, called the ornamentation lattice of $\mathsf{B}$. Special cases of this construction have already proven useful in understanding the structure of three families of posets: operahedron lattices, the affine Tamari lattice, and hypergraphic posets of subhypergraphs of the path hypergraph of an increasing tree. The goal of this paper is to establish the theory of these generalized ornamentations. We examine several natural classes of pointed building sets which recover classical lattices such as the Tamari lattice, the lattice of topologies ordered by coarsening, and the lattice of naturally labeled partial orders. Furthermore, several theoretical directions are explored, including inverse limits and group actions. Notably, this leads to a straightforward construction of inverse limits of Tamari lattices, yielding infinite analogs of the Tamari lattice.

Figures (8)

• Figure 1: An example of a pointed building set. In this example, all pointed sets are pointed at their minimal element. We color the outline of a set with the same color as its pointed element.
• Figure 2: The ornamentation lattice of the pointed building set in \ref{['fig:pointed_building_set_example']}.
• Figure 3: An example of $\sigma$ in $\mathrm{Tam}_3$. Observe that $\sigma$ is not necessarily an ornamentation.
• Figure 4: Illustration of \ref{['thm:tree_duality']}. Top: $\mathcal{O}(\mathsf{B}(D))$. Bottom: $\mathcal{O}(\mathsf{B}(D^\mathrm{op}))^\mathrm{op}$. Each tree is oriented upwards. Bottom is a relabelling of \ref{['fig:ornamentations_example']}.
• Figure 5: Failure of \ref{['thm:tree_duality']} for a directed acyclic graph $D$. Observe that $D^\mathrm{op} \simeq D$ but $\mathcal{O}(\mathsf{B}(D))^\mathrm{op} \not\simeq \mathcal{O}(\mathsf{B}(D^\mathrm{op}))$.

Theorems & Definitions (111)

• Definition 1.1
• Example 1.2
• Definition 1.3
• Theorem 1.4
• Example 3.1
• Example 3.2
• Example 3.3
• Example 3.4
• Example 3.5
• Example 3.6