Extension of the regionally proximal equivalence theorem beyond commutative semigroups

Determine to what extent the conclusions that the regionally proximal relation equals the equicontinuous structure relation, is an equivalence relation, and yields the maximal equicontinuous factor extend to minimal actions of more general semigroups.

Background

The paper proves that for minimal actions of commutative semigroups, the regionally proximal relation is an equivalence relation and coincides with the equicontinuous structure relation. It also notes that analogous conclusions fail for some minimal actions of non-abelian groups.

The authors leave unresolved how broadly their conclusions extend when the acting semigroup is not assumed to be commutative.

References

We do not know to what extent the conclusions of \cref{mainthm_RP_is_eq_relation} hold for more general semigroups $S$.

— The regionally proximal relation for commutative semigroup actions  (2610.10397 - Blahodatna et al., 7 Oct 2026) in Section 1, subsection “Regional proximality and equicontinuity for semigroup actions,” paragraph following Theorem 1.2

Determine the possible relationships between $RP$ and $RP-$. For example, is it true that $RP-$ is always a subset of $RP$? Are there natural necessary and sufficient conditions for $RP = RP-$?

— The regionally proximal relation for commutative semigroup actions  (2610.10397 - Blahodatna et al., 7 Oct 2026) in Section “Open directions,” subsection “Forward and backward regionally proximal behavior”

Let $S$ be an semigroup and $X$ be a minimal $S$-system. If $S$ fixes a Borel probability measure on $X$, is $RP$ an equivalence relation? In particular, if $S$ is left amenable, is $RP$ an equivalence relation?

— The regionally proximal relation for commutative semigroup actions  (2610.10397 - Blahodatna et al., 7 Oct 2026) in Section “Open directions,” subsection “Beyond actions of commutative semigroups”

Let $S$ be a commutative semigroup and $X$ be a minimal $S$-system. Is $RP{[d]}$ an equivalence relation?

— The regionally proximal relation for commutative semigroup actions  (2610.10397 - Blahodatna et al., 7 Oct 2026) in Section “Open directions,” subsection “Higher-order generalizations,” at Question \ref{quest_higher_order_rp_for_semigroups}

Is there a generalization of $RP$ for semigroup actions in line with the one for group actions in which is an equivalence relation under all minimal actions of arbitrary semigroups?

— The regionally proximal relation for commutative semigroup actions  (2610.10397 - Blahodatna et al., 7 Oct 2026) in Section “Open directions,” subsection “Higher-order generalizations,” final question