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
Abbadini M.
2021-01-01
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.File | Dimensione | Formato | |
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