Local finiteness of Cayley graphs for finitely generated multivalued groups

Determine whether, for every finitely generated n-valued group X and any finite generating set S, the Cayley graph Γ_{X,S} defined by vertices X and directed edges u → (u * s)_k for s ∈ S and k ∈ {1,…,n} is locally finite (i.e., each vertex has finitely many incident edges).

Background

The paper defines the Cayley graph Γ_{X,S} for an n-valued group X with a finite generating set S as an oriented graph whose vertices are X and whose directed edges encode the multivalued product: there is an edge from u to each (u * s)_k for s in S and k=1,…,n. Because multivalued products can have repeated outcomes and the graph is directed, local finiteness is not automatic as in the ordinary group case.

After introducing the Set map to handle multi-sets, the authors pose whether the resulting Cayley graph is locally finite. This question probes foundational geometric properties of multivalued groups and whether standard finiteness behavior from group Cayley graphs extends to the multivalued setting.