Equivalence à la Mundici for commutative lattice-ordered monoids
Tóm tắt
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
$$x \mapsto -x$$
. The primitive operations are
$$+$$
,
$$\vee $$
,
$$\wedge $$
, 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
$$\oplus $$
,
$$\odot $$
,
$$\vee $$
,
$$\wedge $$
, 0, 1. A motivating example of MV-monoidal algebra is the negation-free reduct of the standard MV-algebra
$$[0, 1]\subseteq \mathbb {R}$$
. We obtain the original Mundici’s equivalence as a corollary of our main result.
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