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In this paper we study interpretations and equivalences of propositional deductive systems by using a quantale-theoretic approach introduced by Galatos and Tsinakis. Our aim is to provide a general order-theoretic framework which is able to describe and characterize both strong and weak forms of interpretations among propositional deductive systems also in the cases where the systems have different underlying languages.Ackowledgements. I am extremely grateful to Constantine Tsinakis for his participation in the development of this work. In my opinion, his role in the preparation of this paper deserved a coauthorship, which he unfortunately declined.

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