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Coproduct and Amalgamation of Deductive Systems by Means of Ordered Algebras

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Abstract

We propose various methods for combining or amalgamating propositional languages and deductive systems. We make heavy use of quantales and quantale modules in the wake of previous works by the present and other authors. We also describe quite extensively the relationships among the algebraic and order-theoretic constructions and the corresponding ones based on a purely logical approach.

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Notes

  1. Using a different symbol for this action would make the notations much heavier without helping the reading, so we rather preferred to use the same symbol of the product in the quantale, relying on the context and different sets of letters for scalars and “vectors” for the meaning of each of its occurrences. Whenever convenient, we shall also drop it.

  2. Products and coproducts in \(Q\text {-}\mathcal {M}\!\!\, od \) have the same object, namely, the Cartesian product with componentwise operations [30, Proposition 4.2.3].

  3. We recall that \( Fm _\mathcal {L}\) and \( Eq _\mathcal {L}\) can be thought of as the sets of, respectively, \(\{(0,1)\}\)-sequents and \(\{(1,1)\}\)-sequents.

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  1. Departamento de Matemática Instituto de Matemática e Estatística, Universidade Federal da Bahia, Salvador, Bahia, Brazil

    Ciro Russo

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This paper has been awarded the 2021 Newton da Costa Prize for Logic and will be presented at the 2nd World Logic Prizes Contest within the UNILOG 2022 conference, in Crete. This work was supported by the individual travel grant Professor Visitante no Exterior Sênior - Grant No. 88887.477515/2020-00, awarded by the Coordenadoria de Aperfeiçoamento de Pessoal de Nível Superior and the Universidade Federal da Bahia through the CAPES-PrInt UFBA.

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Russo, C. Coproduct and Amalgamation of Deductive Systems by Means of Ordered Algebras. Log. Univers. 16, 355–380 (2022). https://doi.org/10.1007/s11787-022-00303-x

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