Upho lattices II: ways of realizing a core

TL;DR

This paper investigates how many different upho lattices can have a given finite graded lattice as their core, revealing finiteness under certain conditions and unbounded possibilities in others.

Contribution

It introduces the problem of counting realizations of a finite lattice as a core of an upho lattice and provides results on finiteness and unboundedness of these realizations.

Findings

Finite lattices with no nontrivial automorphisms have finitely many upho realizations.

The number of realizations can be unbounded, even for rank-two lattices.

Discussion of an algorithm for listing all realizations.

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.