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Extension Properties and Subdirect Representation in Abstract Algebraic Logic

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Abstract

This paper continues the investigation, started in Lávička and Noguera (Stud Log 105(3): 521–551, 2017), of infinitary propositional logics from the perspective of their algebraic completeness and filter extension properties in abstract algebraic logic. If follows from the Lindenbaum Lemma used in standard proofs of algebraic completeness that, in every finitary logic, (completely) intersection-prime theories form a basis of the closure system of all theories. In this article we consider the open problem of whether these properties can be transferred to lattices of filters over arbitrary algebras of the logic. We show that in general the answer is negative, obtaining a richer hierarchy of pairwise different classes of infinitary logics that we separate with natural examples. As by-products we obtain a characterization of subdirect representation for arbitrary logics, develop a fruitful new notion of natural expansion, and contribute to the understanding of semilinear logics.

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Acknowledgements

We thank the anonymous referees for their suggestions and corrections. Both authors were supported by the grant GA17-04630S of the Czech Science Foundation. Moreover, this project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 689176.

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Authors and Affiliations

  1. Institute of Information Theory and Automation, Czech Academy of Sciences, Pod Vodárenskou Věží 4, 182 08, Prague, Czech Republic

    Tomáš Lávička & Carles Noguera

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  1. Tomáš Lávička
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  2. Carles Noguera
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Correspondence to Carles Noguera.

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Presented by Jacek Malinowski

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Lávička, T., Noguera, C. Extension Properties and Subdirect Representation in Abstract Algebraic Logic. Stud Logica 106, 1065–1095 (2018). https://doi.org/10.1007/s11225-017-9771-7

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