Equivalence à la Mundici for commutative lattice-ordered monoids

Abstract

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.