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