We answer Mundici's problem number 3 (Mundici (2011) [37]): 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.
Are locally finite MV-algebras a variety?
Abbadini M.;Spada L.
2021-01-01
Abstract
We answer Mundici's problem number 3 (Mundici (2011) [37]): 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
