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The poset of maximal tubings of the cycle graph is a lattice

Ben Adenbaum, Emily Barnard, Max Hlavacek, Bryson Kagy, Nathan R. T. Lesnevich, George D. Nasr, Katie Waddle

TL;DR

This work proves that the poset of maximal tubings of the cycle graph, MTub$(C_n)$, is a lattice and moreover semidistributive and congruence uniform. It develops a global order description via $G$-trees, and introduces the Cut map from MTub$(C_n)$ to MTub$(P_n)$ to relate cycle tubings to the path case; fiber structure and tree-move analyses yield joins and meets and a lattice quotient structure. The authors enumerate join-irreducible elements as $(n-1)^2$ and establish a tight correspondence between join and meet irreducibles, enabling a semidistributive proof and the congruence-uniform property through the Fundamental Theorem for semidistributive lattices. These results place MTub$(C_n)$ among well-studied lattice families (like the weak order and Tamari lattice) and open avenues for connections with Cambrian lattices, torsion classes, and shard-intersection orders. The work thus advances understanding of when graph tubings give lattice structures and highlights rich algebraic and geometric facets of cycle-based graph associahedra.

Abstract

The poset of maximal tubings of a graph generalizes several well-known and remarkable partial orders. Notable examples include the weak Bruhat order and the Tamari lattice, posets of maximal tubings for the complete graph and the path graph, respectively. It is an open problem to characterize graphs for which the poset of maximal tubings is a lattice. In this paper, we prove that the poset of maximal tubings for the cycle graph is a lattice, and moreover that it is semidistributive and congruence uniform. As main tools, we characterize all order relations in the poset, and introduce a useful map from maximal tubings of the cycle graph to maximal tubings of the path graph.

The poset of maximal tubings of the cycle graph is a lattice

TL;DR

This work proves that the poset of maximal tubings of the cycle graph, MTub, is a lattice and moreover semidistributive and congruence uniform. It develops a global order description via -trees, and introduces the Cut map from MTub to MTub to relate cycle tubings to the path case; fiber structure and tree-move analyses yield joins and meets and a lattice quotient structure. The authors enumerate join-irreducible elements as and establish a tight correspondence between join and meet irreducibles, enabling a semidistributive proof and the congruence-uniform property through the Fundamental Theorem for semidistributive lattices. These results place MTub among well-studied lattice families (like the weak order and Tamari lattice) and open avenues for connections with Cambrian lattices, torsion classes, and shard-intersection orders. The work thus advances understanding of when graph tubings give lattice structures and highlights rich algebraic and geometric facets of cycle-based graph associahedra.

Abstract

The poset of maximal tubings of a graph generalizes several well-known and remarkable partial orders. Notable examples include the weak Bruhat order and the Tamari lattice, posets of maximal tubings for the complete graph and the path graph, respectively. It is an open problem to characterize graphs for which the poset of maximal tubings is a lattice. In this paper, we prove that the poset of maximal tubings for the cycle graph is a lattice, and moreover that it is semidistributive and congruence uniform. As main tools, we characterize all order relations in the poset, and introduce a useful map from maximal tubings of the cycle graph to maximal tubings of the path graph.

Paper Structure

This paper contains 11 sections, 60 theorems, 55 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Graphs and maximal tubings
  3. G-trees
  4. G-trees for maximal tubings of Pn
  5. G-trees for maximal tubings of Cn
  6. Cut Map
  7. Global Relations in MTub Cycle
  8. Lattice
  9. Join and Meet Irreducible Elements of MTub(Cn)
  10. Semidistributive and Congruence Uniform
  11. Future Work

Key Result

Lemma 2.3

Let $\mathcal{T} \in \mathop{\mathrm{MTub}}\nolimits(G)$. For each nontrivial tube $T \in \mathcal{T}$, there is a unique tube $J \neq T$ where such that $(\mathcal{T} \setminus \{T\})\cup \{J\}$ is also a maximal tubing for $G$.

Figures (7)

  • Figure 1: A general structure for the $G$-tree of an element of $\mathop{\mathrm{MTub}}\nolimits(P_n)$.
  • Figure 2: A tree move on a binary search tree.
  • Figure 3: The tree move on $(x,m)$ in a cyclic binary tree.
  • Figure 4: An example application of the $\mathop{\mathrm{Cut}}\nolimits$ map to a maximal tubing $\mathcal{J}\in\mathop{\mathrm{MTub}}\nolimits(C_n)$.
  • Figure 5: A $G$-tree for $\mathcal{J}\in\mathop{\mathrm{MTub}}\nolimits(C_n)$ and a $G$-tree for $\mathop{\mathrm{Cut}}\nolimits(\mathcal{J})\in\mathop{\mathrm{MTub}}\nolimits(P_n)$. Note that $\mathop{\mathrm{Sew}}\nolimits_{\mathop{\mathrm{Cut}}\nolimits(\mathcal{J})}(9137)=\mathcal{J}$ (see Example \ref{['ex:Cut_map_inverse']}.)
  • ...and 2 more figures

Theorems & Definitions (150)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Corollary 2.5
  • Definition 2.6
  • Remark 2.7
  • Remark 2.8
  • Definition 2.9
  • ...and 140 more