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.

中文翻译：

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局部有限 MV 代数是多种多样的吗？

我们回答 Mundici 的问题 3（Mundici (2011) [37]）：局部有限 MV 代数的范畴是否等价于方程类？我们证明：

1.

局部有限MV-代数的范畴不等价于任何有限变体。

2.

更多的是真的：局部有限 MV 代数的范畴不等价于任何有限排序的有限拟变体。

3.

局部有限MV-代数的范畴等价于无穷变体；最多可数的操作。

4.

局部有限MV-代数的范畴等价于可数排序的有限簇。

我们的证明基于局部有限 MV 代数与 R. Cignoli、EJ Dubuc 和 D. Mundici 的“多重集”范畴之间的对偶性，以及已知的变体和准变体的分类特征。事实上，除了上述对偶性之外，不需要任何 MV 代数知识。