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Projectivity in (bounded) commutative integral residuated lattices

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Abstract

In this paper, we study projective algebras in varieties of (bounded) commutative integral residuated lattices. We make use of a well-established construction in residuated lattices, the ordinal sum, and the order property of divisibility. Via the connection between projective and splitting algebras, we show that the only finite projective algebra in \(\mathsf {{FL}_{ew}}\) is the two-element Boolean algebra. Moreover, we show that several interesting varieties have the property that every finitely presented algebra is projective, such as locally finite varieties of hoops. Furthermore, we show characterization results for finite projective Heyting algebras, and finitely generated projective algebras in locally finite varieties of bounded hoops and BL-algebras. Finally, we connect our results with the algebraic theory of unification.

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Acknowledgements

The authors wish to thank the anonymous referee who suggested an improvement of Lemma 3.1. The authors declare that this work has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant agreement No 890616 awarded to Ugolini.

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Authors and Affiliations

  1. DIISM, Università di Siena, Siena, Italy

    Paolo Aglianò

  2. Artificial Intelligence Research institute (IIIA), CSIC, Bellaterra, Barcelona, Spain

    Sara Ugolini

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  1. Paolo Aglianò
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  2. Sara Ugolini
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Correspondence to Sara Ugolini.

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Presented by N. Galatos.

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Aglianò, P., Ugolini, S. Projectivity in (bounded) commutative integral residuated lattices. Algebra Univers. 84, 2 (2023). https://doi.org/10.1007/s00012-022-00798-x

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