Are locally finite MV-algebras a variety?

Abstract

We answer Mundici's problem number 3 (D. Mundici. Advanced {\L}ukasiewicz calculus. Trends in Logic Vol. 35. Springer 2011, p. 235): Is the category of locally finite MV-algebras equivalent to an equational class? We prove: (i) The category of locally finite MV-algebras is not equivalent to any finitary variety. (ii) More is true: the category of locally finite MV-algebras is not equivalent to any finitely-sorted finitary quasi-variety. (iii) The category of locally finite MV-algebras is equivalent to an infinitary variety; with operations of at most countable arity. (iv) 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 categorical characterisations of varieties and quasi-varieties proved by J. Duskin, J. R. Isbell, F. W. Lawvere and others. In fact no knowledge on MV-algebras is needed, apart from the aforementioned duality.