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On prevarieties of logic

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Abstract

It is proved that every prevariety of algebras is categorically equivalent to a ‘prevariety of logic’, i.e., to the equivalent algebraic semantics of some sentential deductive system. This allows us to show that no nontrivial equation in the language \(\wedge ,\vee ,\circ \) holds in the congruence lattices of all members of every variety of logic, and that being a (pre)variety of logic is not a categorical property.

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Authors and Affiliations

  1. Institute of Computer Science, Academy of Sciences of the Czech Republic, Pod Vodárenskou věží 2, 182 07, Prague 8, Czech Republic

    Tommaso Moraschini

  2. Department of Mathematics and Applied Mathematics, University of Pretoria, Hatfield, Private Bag X20, Pretoria, 0028, South Africa

    James G. Raftery

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  1. Tommaso Moraschini
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  2. James G. Raftery
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Correspondence to James G. Raftery.

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This work received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 689176 (project “Syntax Meets Semantics: Methods, Interactions, and Connections in Substructural logics”). The first author was supported by project CZ.02.2.69/0.0/0.0/17_050/0008361, OPVVV MŠMT, MSCA-IF Lidské zdroje v teoretické informatice. The second author was supported in part by the National Research Foundation (NRF) of South Africa (UID 85407). Both authors thank the University of Pretoria Staff Exchange Bursary Programme and the DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS) (RT18ALG/006) for partially funding the first author’s travel to Pretoria in 2017 and 2018, respectively. Opinions expressed and conclusions arrived at are those of the authors and are not necessarily to be attributed to the CoE-MaSS.

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Moraschini, T., Raftery, J.G. On prevarieties of logic. Algebra Univers. 80, 37 (2019). https://doi.org/10.1007/s00012-019-0611-7

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  • DOI: https://doi.org/10.1007/s00012-019-0611-7

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