Representing partition lattices through FCA

TL;DR

This paper explores the structure of partition lattices using Formal Concept Analysis, providing methods to construct join-irreducible elements and analyzing their growth, which enhances understanding of lattice representations.

Contribution

It introduces a construction method for join-irreducible elements of partition lattices and analyzes the size of their formal contexts, advancing lattice theory and FCA applications.

Findings

Number of join-irreducible elements grows quadratically with n

The formal context has order Theta(n^2) in size

Constructive method for join-irreducibles from smaller lattices

Abstract

We investigate the standard context, denoted by $\mathbb{K}\left(\mathcal{L}_{n}\right)$, of the lattice $\mathcal{L}_{n}$ of partitions of a positive integer $n$ under the dominance order. Motivated by the discrete dynamical model to study integer partitions by Latapy and Duong Phan and by the characterization of the supremum and (infimum) irreducible partitions of $n$ by Brylawski, we show how to construct the join-irreducible elements of $\mathcal{L}_{n+1}$ from $\mathcal{L}_{n}$. We employ this construction to count the number of join-irreducible elements of $\mathcal{L}_{n}$, and show that the number of objects (and attributes) of $\mathbb{K}\left(\mathcal{L}_{n}\right)$ has order $\Theta(n^2)$.