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