Upho lattices I: examples and non-examples of cores

TL;DR

This paper explores the structure and classification of finite upho lattices, focusing on their cores, and identifies which finite graded lattices can or cannot serve as cores, providing new insights into their properties.

Contribution

It introduces the concept of cores of upho lattices, characterizes which finite graded lattices can be cores, and presents both constructive examples and obstructions.

Findings

Many well-studied finite lattices can be realized as cores of upho lattices.

Constructive methods for embedding certain lattices into upho lattices are provided.

Obstructions are identified that prevent some finite lattices from being cores.

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.