# Equivalence \`a la Mundici for commutative lattice-ordered monoids

## TL;DR

This paper generalizes Mundici's equivalence by establishing a categorical equivalence between unital commutative lattice-ordered groups and MV-monoidal algebras, which are negation-free variants of classical structures.

## Contribution

It introduces unital commutative lattice-ordered groups and MV-monoidal algebras, extending Mundici's equivalence to a broader class of algebraic structures.

## Key findings

- Established categorical equivalence between unital commutative lattice-ordered groups and MV-monoidal algebras.
- Provided concrete examples including the real numbers and negation-free reducts of standard MV-algebras.
- Revealed that Mundici's original equivalence is a special case of this broader framework.

## 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 groups is equivalent to the category of MV-monoidal algebras. Roughly speaking, the structures we call unital commutative lattice-ordered groups are unital Abelian lattice-ordered groups without the unary operation $x \mapsto -x$. The primitive operations are $+$, $\lor$, $\land$, $0$, $1$, $-1$. A prime example of these structures is $\mathbb{R}$, with the obvious interpretation of the operations. Analogously, MV-monoidal algebras are MV-algebras without the negation $x \mapsto \lnot x$. The primitive operations are $\oplus$, $\odot$, $\lor$, $\land$, $0$, $1$. A motivating example of MV-monoidal algebra is the negation-free reduct of the standard MV-algebra $[0, 1] \subseteq \mathbb{R}$. We obtain the original Mundici's equivalence as a corollary of our main result.