Abstract
We answer Mundici’s problem number 3 (D. Mundici, Advanced Łukasiewicz Calculus and MV-Algebras, Trends in Logic—Studia Logica Library, vol. 35, Springer, Dordrecht (2011), p. 235): Is the category of locally finite MV-algebras equivalent to an equational class? We prove:
1. The category of locally finite MV-algebras is not equivalent to any finitary variety.
2. More is true: the category of locally finite MV-algebras is not equivalent to any finitely-sorted finitary quasi-variety.
3. The category of locally finite MV-algebras is equivalent to an infinitary variety; with operations of at most countable arity.
4. The category of locally finite MV-algebras is equivalent to a countably-sorted finitary variety.
Our proofs rest upon the duality between locally finite MV-algebras and the category of “multisets” by R. Cignoli, E.J. Dubuc and D. Mundici, and known categorical characterisations of varieties and quasi-varieties. In fact, no knowledge of MV-algebras is needed, apart from the aforementioned duality.
1. The category of locally finite MV-algebras is not equivalent to any finitary variety.
2. More is true: the category of locally finite MV-algebras is not equivalent to any finitely-sorted finitary quasi-variety.
3. The category of locally finite MV-algebras is equivalent to an infinitary variety; with operations of at most countable arity.
4. The category of locally finite MV-algebras is equivalent to a countably-sorted finitary variety.
Our proofs rest upon the duality between locally finite MV-algebras and the category of “multisets” by R. Cignoli, E.J. Dubuc and D. Mundici, and known categorical characterisations of varieties and quasi-varieties. In fact, no knowledge of MV-algebras is needed, apart from the aforementioned duality.
| Original language | English |
|---|---|
| Article number | 106858 |
| Number of pages | 28 |
| Journal | Journal of Pure and Applied Algebra |
| Volume | 226 |
| Issue number | 4 |
| Early online date | 28 Jul 2021 |
| DOIs | |
| Publication status | Published - Apr 2022 |
Bibliographical note
Acknowledgments:The research of both authors was supported by the Italian Ministry of University and Research through the PRIN project n. 20173WKCM5 Theory and applications of resource sensitive logics.