Single-restriction characterization of M(O) for disklike transfer systems

Establish that for every disklike G‑transfer system O, the maximal compatible transfer system M(O) consists exactly of those transfers e in O such that for every non‑reflexive restriction r of e in O, the restriction is a compatibility success; concretely, for each restriction r: K∩J → J of e: K → H, if K∩J → K lies in O then J → H also lies in O, so that M(O) = { e ∈ O | ∀ r < e, r is a compatibility success for e }.

Background

The paper develops formulas for the maximal compatible transfer system M(O) associated to a given transfer system O and proves simplified characterizations when O is disklike. In particular, Theorem 4.4 shows a recursive criterion based on covered relations, and an efficient algorithm follows for disklike cases.

The authors conjecture a sharper simplification: for disklike O it should suffice to check that every non‑reflexive restriction of a transfer e is a compatibility success, eliminating the need for recursive checks and cover‑relation reductions. They provide experimental support and verify the claim for various families of groups and transfer systems.

References

Conjecture [\cref{conj:maximal compatible disk-like via single restriction}] Let $O$ be a disklike $G$-transfer system, then \begin{align*}

1{O} = { e \in O \, | \, \forall r < e, r <S e }, \end{align*}

where $r<e$ means, that in the poset of restrictions in $O$, the transfer $e$ restricts into $r$, and $r<S e$ means that this restriction satisfies the compatibility condition (\cref{def:compatibility successes and failures}).

— Maximal compatibility of disklike $G$-transfer systems  (2604.00335 - DeMark et al., 1 Apr 2026) in Subsection: Disklike Conjectured Characterization