AbstractWe 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 $$x \mapsto -x$$
x
↦
-
x
. The primitive operations are $$+$$
+
, $$\vee $$
∨
, $$\wedge $$
∧
, 0, 1, $$-1$$
-
1
. A prime example of these structures is $$\mathbb {R}$$
R
, with the obvious interpretation of the operations. Analogously, MV-monoidal algebras are MV-algebras without the negation $$x \mapsto \lnot x$$
x
↦
¬
x
. The primitive operations are $$\oplus $$
⊕
, $$\odot $$
⊙
, $$\vee $$
∨
, $$\wedge $$
∧
, 0, 1. A motivating example of MV-monoidal algebra is the negation-free reduct of the standard MV-algebra $$[0, 1]\subseteq \mathbb {R}$$
[
0
,
1
]
⊆
R
. We obtain the original Mundici’s equivalence as a corollary of our main result.
MV-algebras and Partially Cyclically Ordered Groups
