Formal Concept Analysis and Homotopical Combinatorics

TL;DR

This paper explores the structure of finite lattices through Formal Concept Analysis, describing transfer systems with group actions, and provides computational tools and bounds for transfer systems in finite groups.

Contribution

It explicitly characterizes transfer systems in finite lattices with group actions using Formal Concept Analysis, enabling new computations and bounds.

Findings

Explicit description of transfer systems for finite lattices with group actions

New bounds on the number of transfer systems for certain finite groups

Provision of computer code for applying these techniques

Abstract

Formal Concept Analysis makes the fundamental observation that any finite lattice $(L, \leq)$ is determined up to isomorphism by the restriction of the relation ${\leq} \subseteq L \times L$ to the set $J(L) \times M(L)$, where $J(L)$ is the set of join-irreducible elements of $L$ and $M(L)$ is the set of meet-irreducible elements of $L$. For any finite lattice $L$ equipped with the action of a finite group $G$, we explicitly describe this restricted relation for the lattice of transfer systems $\mathsf{Tr}(L)$ in terms of $L$ only. We apply this to give new computations of the number of transfer systems for certain finite groups, and to produce bounds on the number of transfer systems on certain families of abelian finite groups. We also provide computer code to enable other researchers' use of these techniques.