The dual adjunction between MV-algebras and Tychonoff spaces
This interesting and well-written paper is concerned with MV-algebras. These structures are the many-valued equivalents of Boolean algebras for classical, two-valued logic. They can be thought of as non-idempotent Boolean algebras. The most important example of an MV-algebra is the real interval \([0,1]\), where the negation of \(x\) is \(1-x\) and the sum of \(x\) and \(y\) is \(\min(1,x+y)\). The classical Stone duality between Boolean algebras and Stonean topological spaces can be extended to MV-algebras, but only partially. However, in analogy with algebraic geometry, the paper describes in detail a category-theoretic dual adjunction between the MV-algebras (seen as algebraic objects) and the subsets of the Cartesian powers of \([0,1]\) (seen as geometric objects). Like every adjunction, this one specializes to a dual equivalence between the fixpoints, which turn out to be the semisimple MV-algebras and the closed subsets of the powers of \([0,1]\). This gives an interesting Nullstellensatz for MV-algebras. Finally, when MV-algebras are finitely presented, the geometric counterpart is given by rational polyhedra (as the authors point out, the finitely presented case was already known, but the approach is original).
scientific article; zbMATH DE number 6088860
scientific article