The Formal Context of Saturated Transfer Systems on Finite Abelian Groups

TL;DR

This work develops a Formal Concept Analysis–driven framework to study saturated transfer systems on finite groups, focusing on finite modular lattices and, in particular, Sub$(G)$ for abelian $G$. It identifies a complete description of the reduced formal context for Sat$(L)$ (via join-irreducible and meet-irreducible elements) and proves that, for finite modular $L$, the irreducible saturated transfer systems are precisely $ig floor x o yig floor$ for covering relations and $igracebot o yigraceboxslash$ for $y eqot$, with a simple incidence criterion. This yields the full reconstruction of Sat$(L)$ from its reduced context and enables explicit enumeration, including the total number of saturated transfer systems on $ ext{Sub}(C_5^3)$, which is $13{,}784{,}538{,}270{,}571$. Additionally, the paper develops density results for Sat$( ext{Sub}(C_p^n))$, giving Gaussian-binomial–based counts of meet-irreducibles, a recurrence for the number of subspaces, and asymptotic density behavior as $n$ or $p$ grow, providing insight into the relative complexity of these lattices in families of elementary abelian $p$-groups.

Abstract

We describe the reduced formal context of the lattice of saturated transfer systems on a finite abelian group. As an application, we compute that there are 13,784,538,270,571 saturated transfer systems on the elementary abelian group $C_5^3$.

Figures (1)

• Figure 1: The reduced formal context for the lattice $\mathop{\mathrm{\mathsf{Sat}}}\nolimits(\mathop{\mathrm{\mathsf{Sub}}}\nolimits(C_5^3))$. A black pixel represents a $0$ in the binary matrix and a white pixel represents a $1$.

Theorems & Definitions (49)

• Theorem A
• Theorem B
• Theorem C
• Definition 2.1
• Definition 2.2
• Proposition 2.3
• Proposition 2.4
• Definition 2.5
• Definition 2.6
• Definition 2.7