Equality of forward and backward higher-order regional proximality

Determine under what conditions the forward higher-order relation $RP^{[d]}$ and a backward variant $(RP^{[d]})^-$ coincide for commutative semigroup actions.

Background

The paper suggests introducing a backward higher-order regional proximal relation in analogy with the backward relation RP−RP^- studied for order two. Although the paper proves equality of the forward and backward order-two relations for minimal commutative semigroup actions, it does not establish an analogous result at higher order.

The unresolved issue is to identify assumptions under which the two higher-order variants agree.

References

A natural first question: under what conditions are the forward and backward variants the same?

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