Upho lattices I: examples and non-examples of cores

TL;DR

This work studies upho lattices—infinite posets whose principal filters reproduce the whole structure—and focuses on finite-type $\mathbb{N}$-graded lattices. A key tool is the core $L=[\hat{0},s_1\vee\cdots\vee s_r]$ of an upho lattice, which determines the rank generating function $F(\mathcal{L};x)$ via $F(\mathcal{L};x)=\chi^*(L;x)^{-1}$. The authors develop a positive construction: any member of a uniform sequence of supersolvable geometric lattices is the core of some upho lattice; this yields cores for Boolean, subspace, partition, Type B partitions, and Dowling lattices, among others, through both combinatorial and monoid/Coxeter-based methods. They also identify obstructions—via reciprocal characteristic polynomials and structural self-similarity requirements—that prevent many lattices from being cores (e.g., cross polytopes, hypercubes, cycle-bond lattices, and flats of uniform matroids with certain parameters). Altogether, the paper maps the landscape of cores for upho lattices, providing both broad construction techniques and concrete non-core instances, and lays groundwork for refined classification and further subvarieties such as distributive or modular upho lattices."

Abstract

A poset is called upper homogeneous, or "upho," if every principal order filter of the poset is isomorphic to the whole poset. We study (finite type $\mathbb{N}$-graded) upho lattices, with an eye towards their classification. Any upho lattice has associated to it a finite graded lattice called its core, which determines its rank generating function. We investigate which finite graded lattices arise as cores of upho lattices, providing both positive and negative results. On the one hand, we show that many well-studied finite lattices do arise as cores, and we present combinatorial and algebraic constructions of the upho lattices into which they embed. On the other hand, we show there are obstructions which prevent many finite lattices from being cores.

Figures (11)

• Figure 1: Partitions of sets of the form $\{1,2,\ldots,n\}$ into $2$ blocks, ordered by refinement. This is an upho lattice with core $\Pi_3$.
• Figure 2: The dual braid monoid $\langle a,b,c\mid ab=bc=ca\rangle$ associated to the symmetric group $S_3$. This is an upho lattice with core the noncrossing partition lattice of $S_3$.
• Figure 3: The "trimmed" partition lattice $\Pi^{(2)}_4$.
• Figure 4: The monoid $M=\langle a,b\mid ba=aa\rangle$ from the case $r=2$ of \ref{['thm:rank_two']}, an upho lattice with core $M_2$.
• Figure 5: The weak order of the symmetric group $S_3$.

Theorems & Definitions (82)

• Theorem 1.3
• Lemma 1.4
• Example 2.1
• Example 2.2
• Example 2.3
• Example 2.4
• Example 2.5
• Example 2.6
• Example 2.7
• Example 2.8