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Perfect MV-algebras are categorically equivalent to Abelian $$l$$-groups. (English) Zbl 0812.06010
An MV-algebra is an abelian monoid $$(B,\oplus,0)$$ with an operation $$*$$ such that $$x^{**} = x$$, $$x \oplus 0^* = 0^*$$ and $$(x^* \oplus y)^* \oplus y = (y^* \oplus x)^* \oplus x$$. One usually also defines $$x \bullet y = (x^* \oplus y^*)^*$$ and $$1 = 0^*$$. For any abelian lattice-group $$G$$ with strong unit $$u$$, let $$\Gamma(G,u) = [0,u] = \{x \in G \mid 0 \leq x \leq u\}$$ equipped with the operations $$x^* = u - x$$, $$x \oplus y = (x + y) \wedge u$$. Further, for every morphism $$f : (G,u) \to (G',u')$$ let $$\Gamma(f)$$ be the restriction of $$f$$ to $$[0,1]$$. Then, as proved by the present reviewer in his paper “Interpretation of AF $$C^*$$-algebras in Łukasiewicz sentential calculus” [J. Funct. Anal. 65, 15-63 (1986; Zbl 0597.46059)], $$\Gamma$$ is a categorical equivalence between abelian lattice-groups with strong units, and MV- algebras. The order $$\text{ord }x$$ of an element $$x$$ in an MV-algebra $$B$$ is the smallest integer $$n$$ such that $$1 = x \oplus x \oplus \dots \oplus x$$ ($$n$$ times). If no such $$n$$ exists then $$\text{ord }x = \infty$$. An MV-algebra $$B$$ is perfect iff for every $$x \in B$$ exactly one of $$x$$ and $$x^*$$ is of finite order. For every abelian lattice-group $$G$$, let $$G' = \mathbb{Z} \times G$$, where $$\times$$ denotes lexicographic product, equipped with the strong unit $$u' = (1,0)$$. Then $$\Gamma(G',u')$$ is a perfect MV- algebra. The authors prove that the functor $$G \mapsto \Gamma(G',u')$$ is a categorical equivalence between abelian lattice-groups and perfect MV- algebras; they also show that every perfect MV-algebra is in the variety $$V(C)$$ generated by Chang’s algebra $$C = \Gamma(\mathbb{Z} \times \mathbb{Z},(1,0))$$, and that subdirect products of totally ordered perfect MV- algebras exhaust the variety $$\text{BP}_ 0$$ of bipartite MV-algebras, those MV-algebras $$B$$ for which for every maximal ideal $$M$$, $$B = M \cup M^*$$. As a consequence, $$V(C)$$ coincides with $$\text{BP}_ 0$$, and both varieties are characterized by the single identity $$(x \oplus x) \bullet (x \oplus x) = (x \bullet x) \oplus (x \bullet x)$$. Finally, the authors describe coproducts of perfect MV-algebras, and prove that the variety $$V(C)$$ has the amalgamation property.
Reviewer: D.Mundici (Milano)

##### MSC:
 06F20 Ordered abelian groups, Riesz groups, ordered linear spaces 06D30 De Morgan algebras, Łukasiewicz algebras (lattice-theoretic aspects) 06F99 Ordered structures 18B99 Special categories
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