Lattices from Pointed Building Sets: Generalized Ornamentation Lattices

TL;DR

This paper introduces pointed building sets and their ornamentation lattices, unifying and generalizing classical lattices like Tamari and topology lattices, and explores their theoretical properties including inverse limits and group actions.

Contribution

It develops the theory of pointed building sets and ornamentation lattices, connecting them to classical and new lattice structures, and constructs infinite Tamari lattices via inverse limits.

Findings

Unified framework for classical lattices via pointed building sets

Construction of infinite Tamari lattices through inverse limits

Exploration of group actions on ornamentation lattices

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.