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Adam Přenosil's article "Cut elimination, identity elimination, and interpolation in super-Belnap logics" contains this proposition:

Proposition 3.2. Each sequent is equivalent in the Gentzen relation $\mathrm{G}\mathcal{B}$ to a finite set of atomic sequents. Each finite set of sequents is equivalent in the Gentzen relation $\mathrm{G}\mathcal{B}$ to a single sequent of the form $\emptyset \rhd \varphi$."

The article includes some other preliminaries but does not define equivalence of (sets of) sequents. I looked for a definition in Negri & von Plato but came up empty. My guess is that two sets of sequents are equivalent if all those in the first set can be derived in the calculus in question by assuming all those in the second and vice versa. Can anyone confirm this guess or supply the correct definition?

jdonland
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    Correct; see pag.5 "We say that two sets of formulas Γ and ∆ are equivalent in L in case Γ ⊢ δ for each δ ∈ ∆ and ∆ ⊢ γ for each γ ∈ Γ, and likewise for sequents." – Mauro ALLEGRANZA Feb 22 '22 at 10:36
  • @MauroALLEGRANZA Serves me right for not reading carefully enough. – jdonland Mar 02 '22 at 21:04

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