Perfect MV-algebras are categorically equivalent to Abelian \(l\)-groups
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.
scientific article; zbMATH DE number 683202
scientific article