Recurrence Sets for Partial Inverse Semigroup Actions and Related Structures

The paper proves Theorem 6.8 and Proposition 6.9, establishing the bijection δ̇σ between bisection classes and the source fiber and the identities δ̇σ[B(θ)Nσ]=Ξ̃Nσ and B(θ)NM=∆(Ξ̃NM), with ∆σ=δ−1σ, using the standard properties of étale groupoids and bisections. The candidate solution reproduces the same argument with slightly more detailed justifications (e.g., products with idempotents and local bisection neighborhoods), aligning step-for-step with the paper’s statements and definitions. No logical conflicts were found, and both are correct with essentially the same proof strategy .