Cu-structure of dynamical quotients of G-Cu-semigroups

Determine whether every orbit quotient S/G of a G-Cu-semigroup (S,η) is itself a Cu-semigroup; equivalently, ascertain whether there exists a G-Cu-semigroup (S,η) whose orbit quotient S/G fails to satisfy the Cu-axioms (O1–O4), or prove that such counterexamples cannot occur.

Background

Section 9 develops categorical properties of the Cu-completion and its interaction with group actions, introducing the dynamical completion γ∘ω_G and its natural isomorphisms. After establishing compatibility results between W- and Cu-constructions, the authors explicitly note their lack of a counterexample showing that taking the orbit quotient of a G-Cu-semigroup could leave the Cu category.

This raises a concrete structural question about whether the Cu axioms are preserved under forming dynamical quotients for G-Cu-semigroups. Resolving this would clarify the robustness of Cu-structure under group actions and quotients.