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Amalgamation Property for Some Varieties of BL-Algebras Generated by One Finite Set of BL-Chains with Finitely Many Components

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Relational and Algebraic Methods in Computer Science (RAMiCS 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13896))

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Abstract

BL-algebras are the algebraic semantics of Basic logic BL, the logic of all continuous t-norms and their residua. In a previous work, we provided the classification of the amalgamation property (AP) for the varieties of BL-algebras generated by one BL-chain with finitely many components. As an open problem, we left the analysis of the AP for varieties of BL-algebras generated by one finite set of BL-chains with finitely many components. In this paper we provide a partial solution to this problem. We provide a classification of the AP for the varieties of BL-algebras generated by one finite set of BL-chains with finitely many components, which are either cancellative hoops or finite Wajsberg hoops. We also discuss the difficulties to generalize this approach to the more general case.

M. Bianchi—Independent Researcher.

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Notes

  1. 1.

    The relation \(x\le y\) iff \(x\Rightarrow y=1\) equips \(\bigoplus _{i\in I} \mathcal {A}_i\) with a total order. For every \(x\in A_i,y\in A_j\), \(x\le y\) iff \(x<1\) and \(i<j\) or \(i=j\) and \(x\le _i y\).

  2. 2.

    Note that every non-trivial totally ordered cancellative hoop \(\mathcal {A}\) does not have rank, since \(\mathcal {A}/Rad(\mathcal {A})\) is an infinite cancellative hoop.

  3. 3.

    The assumption that \(Ch(\mathcal {A})\) does not contain trivial chains is essential. Indeed, if \(\mathcal {A}\) is non-trivial, then \(\textbf{ISP}_u(\mathcal {A})\) does not contain trivial algebras.

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Acknowledgements

This work was partially supported by Istituto Nazionale di Alta Matematica (Indam).

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Authors and Affiliations

  1. Department of Computer Science, Università degli Studi di Milano, Via Celoria 18, 20133, Milan, Italy

    Stefano Aguzzoli

  2. Milan, Italy

    Matteo Bianchi

Authors
  1. Stefano Aguzzoli
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  2. Matteo Bianchi
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Correspondence to Matteo Bianchi .

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Editors and Affiliations

  1. Deutsches Zentrum für Luft- und Raumfahrt, Augsburg, Germany

    Roland Glück

  2. Aix-Marseille University, Marseille, France

    Luigi Santocanale

  3. Brock University, St Catharines, ON, Canada

    Michael Winter

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Aguzzoli, S., Bianchi, M. (2023). Amalgamation Property for Some Varieties of BL-Algebras Generated by One Finite Set of BL-Chains with Finitely Many Components. In: Glück, R., Santocanale, L., Winter, M. (eds) Relational and Algebraic Methods in Computer Science. RAMiCS 2023. Lecture Notes in Computer Science, vol 13896. Springer, Cham. https://doi.org/10.1007/978-3-031-28083-2_1

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  • DOI: https://doi.org/10.1007/978-3-031-28083-2_1

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  • Online ISBN: 978-3-031-28083-2

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