Upho lattices II: ways of realizing a core

TL;DR

This work investigates how many upho lattices can realize a given finite graded lattice as its core, proving finiteness when the core has no nontrivial automorphisms while showing the number of realizations can be unbounded, even for rank-two cores. It leverages the deep connection between upho posets and left-cancellative, homogeneously generated monoids to establish colorability and finiteness results, and to construct vast families of upho lattices with a fixed rank-two core $M_n$ via monoid presentations $M(f)$. The rank-two case is developed in detail, with explicit constructions $\mathcal{D}_n$ and $\mathcal{F}_n$, a general monoid family $M(f)$ yielding $\mathcal{L}(f)$, and a lower bound $\kappa(M_n) \ge p(n)$ determined by partitions of $n$ (with a Fibonacci-based refinement). The paper concludes with a speculative algorithmic framework for listing all realizations, highlighting conjectures about pre-upho colorings and their limitations, and pointing to potential use of Garside theory to advance the methodology.

Abstract

A poset is called upper homogeneous, or "upho," if all of its principal order filters are isomorphic to the whole poset. In previous work of the first author, it was shown that each (finite-type N-graded) upho lattice has associated to it a finite graded lattice, called its core, which determines the rank generating function of the upho lattice. In that prior work the question of which finite graded lattices arise as cores was explored. Here, we study the question of in how many different ways a given finite graded lattice can be realized as the core of an upho lattice. We show that if the finite lattice has no nontrivial automorphisms, then it is the core of finitely many upho lattices. We also show that the number of ways a finite lattice can be realized as a core is unbounded, even when restricting to rank-two lattices. We end with a discussion of a potential algorithm for listing all the ways to realize a given finite lattice as a core.

Figures (7)

• Figure 1: Two different upho lattices with core $B_2$.
• Figure 2: Two different colored upho lattices with core $B_2$, as in \ref{['ex:bool_upho1']} and \ref{['ex:bool_upho2']}. Also, compare \ref{['fig:b2_ex']}.
• Figure 3: The lattice $L$ from \ref{['ex:auto_still_finite']} which has $\kappa(L)<\infty$ even though it has a nontrivial automorphism.
• Figure 4: On the left, the dominating vertex construction of an upho lattice $\mathcal{D}_3$ with core $M_3$; and on the right, the flip construction of an upho lattice $\mathcal{F}_3$ with core $M_3$.
• Figure 5: Inputting the coloring of the lattice $L$ on the left into \ref{['conj:precoloring_monoid']} yields an upho lattice $\mathcal{L}$ whose core, depicted on the right, is bigger than $L$. Here $M=\langle a,b,c\mid aa=ba, aaa=caa\rangle$.

Theorems & Definitions (49)

• Remark 2.1
• Theorem 2.2: hopkins2022note
• Corollary 2.3: hopkins2022note
• Lemma 2.4: gao2020upho; see also hopkins2024upho1fu2024upho
• Lemma 3.1: fu2024upho
• proof
• Remark 3.2
• Lemma 3.3
• proof
• Corollary 3.4