Classification of join-irreducibles in saturated transfer-system lattices

Classify the join-irreducible elements of Sat(Sub(G)), the lattice of saturated transfer systems on the subgroup lattice of a finite group G, beyond the abelian case; in particular, provide a complete characterization of which saturated transfer systems generated by a single cover relation are join-irreducible.

Background

The paper studies specialized subclasses of transfer systems important in equivariant homotopy theory, especially saturated and cosaturated lattices. It proves that a join-irreducible element of Sat(Sub(G)) is generated by a single cover relation (Proposition 7.*), but also shows that not every cover relation yields a join-irreducible saturated transfer system, indicating that further structure is required for a full classification.

The authors indicate that the overall classification problem will be pursued in future work, and note that the abelian case will be solved in forthcoming work, leaving the general (non-abelian) case open.