Abstract
BL-algebras are the algebraic semantics for Hájek’s Basic Logic BL, the logic of all continuous t-norms and their residua. Every BL-chain can be decomposed (up to isomorphism) as an ordinal sum of non-trivial Wajsberg hoops - called components - with the first bounded. In this paper we study the amalgamation property for the varieties of BL-algebras generated by one BL-chain with finitely many components.
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Notes
- 1.
\(\varGamma \) establishes a categorical equivalence between abelian l-groups with a strong order unit \((\mathcal {G},u)\) and MV-algebras, by equipping the interval [0, u] of \(\mathcal {G}\) with MV-algebraic operations obtained by truncation of the group ones. On arrows \(\varGamma \) acts by restriction.
- 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.
To use this proof strategy is essential that \(k\ge 4\).
- 4.
To use this proof strategy it is essential that \(h\ge 2\). For every \(l(C_i)\) (\(m(D_i)\), respectively), the elements are ordered as in the chain \(l(\mathcal {C}_i)\) (\(m(\mathcal {D}_i)\), respectively).
- 5.
Here \(x\Leftrightarrow y\) stands for \((x\Rightarrow y)*(y\Rightarrow x)\). Moreover \(x\uplus y{\mathop {=}\limits ^{\text {def}}}(x\Rightarrow (x*y))\Rightarrow y\), whilst nx is defined inductively by \(0x=0\) and \(n(x)=(n-1)x\uplus x\).
- 6.
The assumption that \(Ch(\mathcal {A})\) does not contain trivial chains is essential. Indeed, if \(\mathcal {A}\) is non-trivial, then \(\mathbf {ISP}_u(\mathcal {A})\) does not contain trivial algebras.
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Aguzzoli, S., Bianchi, M. (2021). Amalgamation Property for Varieties of BL-algebras Generated by One Chain with Finitely Many Components. In: Fahrenberg, U., Gehrke, M., Santocanale, L., Winter, M. (eds) Relational and Algebraic Methods in Computer Science. RAMiCS 2021. Lecture Notes in Computer Science(), vol 13027. Springer, Cham. https://doi.org/10.1007/978-3-030-88701-8_1
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