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Implicative twist-structures

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Abstract

The twist-structure construction is used to represent algebras related to non-classical logics (e.g., Nelson algebras, bilattices) as a special kind of power of better-known algebraic structures (distributive lattices, Heyting algebras). We study a specific type of twist-structure (called implicative twist-structure) obtained as a power of a generalized Boolean algebra, focusing on the implication-negation fragment of the usual algebraic language of twist-structures. We prove that implicative twist-structures form a variety which is semisimple, congruence-distributive, finitely generated, and has equationally definable principal congruences. We characterize the congruences of each algebra in the variety in terms of the congruences of the associated generalized Boolean algebra. We classify and axiomatize the subvarieties of implicative twist-structures. We define a corresponding logic and prove that it is algebraizable with respect to our variety.

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Authors and Affiliations

  1. Delft University of Technology, 2600 AA, Delft, The Netherlands

    Umberto Rivieccio

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  1. Umberto Rivieccio
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Correspondence to Umberto Rivieccio.

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Presented by J. Raftery.

The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Unions Seventh Framework Programme (FP7/2007-2013) under REA grant agreement PIEF-GA-2010-272737 and from Vidi grant 016.138.314 of the Netherlands Organization for Scientific Research (NWO).

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Rivieccio, U. Implicative twist-structures. Algebra Univers. 71, 155–186 (2014). https://doi.org/10.1007/s00012-014-0272-5

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