Equivalence à la Mundici for commutative lattice-ordered monoids

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 +, ∨, ∧, 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 ⊕, ⊙, ∨, ∧, 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.