We provide a generalization of Mundici’s equivalence between unital Abelian lattice-ordered groups and MV-algebras: the category of unital commutative lattice-ordered monoids is equivalent to the category of MV-monoidal algebras. Roughly speaking, unital commutative lattice-ordered monoids are unital Abelian lattice-ordered groups without the unary operation x →−x. The primitive operations are +, v, ^, 0, 1, −1. A prime example of these structures is R, with the obvious interpretation of the operations. Analogously, MV-monoidal algebras are MV-algebras without the negation x → ¬x. The primitive operations are +, *, v, ^, 0, 1. A motivating example of MV-monoidal algebra is the negation-free reduct of the standard MV-algebra [0, 1] ⊆ R. We obtain the original Mundici’s equivalence as a corollary of our main result.

### Equivalence {`a} la mundici for commutative lattice-ordered monoids

#### Abstract

We provide a generalization of Mundici’s equivalence between unital Abelian lattice-ordered groups and MV-algebras: the category of unital commutative lattice-ordered monoids is equivalent to the category of MV-monoidal algebras. Roughly speaking, unital commutative lattice-ordered monoids are unital Abelian lattice-ordered groups without the unary operation x →−x. The primitive operations are +, v, ^, 0, 1, −1. A prime example of these structures is R, with the obvious interpretation of the operations. Analogously, MV-monoidal algebras are MV-algebras without the negation x → ¬x. The primitive operations are +, *, v, ^, 0, 1. A motivating example of MV-monoidal algebra is the negation-free reduct of the standard MV-algebra [0, 1] ⊆ R. We obtain the original Mundici’s equivalence as a corollary of our main result.
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2021
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11386/4774792`