Coproduct MV-algebras, nonstandard reals, and Riesz spaces
Introduced by C. C. Chang, MV-algebras are the Lindenbaum algebras of the infinite-valued calculus of Łukasiewicz. They can be defined by a small number of simple equations, capturing the properties of the real unit interval [0,1] equipped with truncated addition and rotation around 1/2. 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)], MV-algebras are categorially equivalent to abelian lattice-ordered groups with strong unit. Suppose \(A\) is an MV-algebra having a unique maximal ideal \(J\). Thus, as a consequence of Hölder's theorem, the quotient \(A/J\) is isomorphic to a subalgebra of the MV-algebra [0,1]. Let \(B\) be a totally ordered MV-algebra, and \(\text{Rad} B\) denote the unique maximal ideal of \(B\). Let \((\text{Rad} B)\) denote the subalgebra of \(B\) generated by \(\text{Rad} B\). The authors use nonstandard real numbers to give concrete representations of those totally ordered MV-algebras that are isomorphic to the coproduct of \(B/ \text{Rad} B\) and \((\text{Rad} B)\). For more information on MV-coproducts see the reviewer's paper ``Free products in the category of abelian \(\ell\)-groups with strong unit'' [J. Algebra 113, 89-109 (1988; Zbl 0658.06010)]. As an application of their representation, the authors discuss the MV-algebraic counterpart of vector lattices.
scientific article; zbMATH DE number 954774