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Codensity for transfer-system contexts of arbitrary finite abelian groups

Determine a general explicit formula for the codensity ρ(Tr(Sub(G))) of the reduced formal context associated to the lattice of transfer systems on Sub(G) for an arbitrary finite abelian group G, extending the results beyond the cyclic and elementary abelian cases.

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Background

The authors compute codensity formulas for two major families of abelian groups: arbitrary cyclic groups and elementary abelian p-groups. These results enable bounds on the size of the lattice of transfer systems via FCA tools.

However, they note that a general result for arbitrary abelian groups remains out of reach because the lattice of transfer systems does not behave well under products: Tr(Sub(G×H)) does not decompose in a manner directly related to Tr(Sub(G)) and Tr(Sub(H)). This obstruction leaves the codensity for general abelian groups unresolved.

References

While between this section and \cref{sec:cyclic} we have computed the (co)density for both families of abelian groups, we are unable to provide a result for an arbitrary abelian group. This is due to the fact that $Tr(Sub(G \times H))$ bears little resemblance to $Tr(Sub(G))$ or $Tr(Sub(H))$ in general.

— Formal Concept Analysis and Homotopical Combinatorics (2507.14068 - Balchin et al., 18 Jul 2025) in Remark, Section 6 (The Reduced Context for Elementary Abelian p-Groups)