Irreducibility and compositional structure of multivariate Alexander quandles

Determine whether irreducible finite multivariate Alexander quandles exist that do not arise by composition of quandles of lower order, and establish whether an analogous compositional structure exists for infinite quandles, particularly in relation to the fundamental link quandle.

Background

All finite multivariate Alexander quandles constructed in the paper decompose into lower-order quandles along diagonal blocks, and their associated quivers reflect this compositional structure.

The authors leave unresolved whether this observed decomposition is universal. They ask both whether genuinely irreducible finite examples exist and whether an analogous decomposition theory persists for infinite quandles, where it could have implications for the fundamental link quandle.

References

This raises two interesting questions: (i) Do there also exist "irreducible" finite multivariate Alexander quandles, which do not result from compositions of lower order quandles? and, (ii) In the case of infinite quandles, does a counterpart to such a compositional structure exist? The latter would be directly relevant to the fundamental link quandle.

— Multivariate Quandles as Groupoid Invariants  (2609.30262 - Arsiwalla et al., 24 Sep 2026) in Section 8, Conclusions and Discussion