The dual adjunction between MV-algebras and Tychonoff spaces
type: Item
instance of: scholarly article
title: The dual adjunction between MV-algebras and Tychonoff spaces
zbMATH Open document ID: 1252.06006
author: Vincenzo Marra
author: Luca Spada
published in: Studia Logica
publication date: 2012-10-01T00:00:00Z
review text: 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).
reviewed by: Giacomo Lenzi
Mathematics Subject Classification ID: 06D35
Mathematics Subject Classification ID: 18A40
Mathematics Subject Classification ID: 54D15
zbMATH DE Number: 6088860
zbMATH Keywords: Łukasiewicz logic
zbMATH Keywords: MV-algebras
zbMATH Keywords: adjunction
zbMATH Keywords: categorical equivalence
zbMATH Keywords: duality
zbMATH Keywords: Tychonoff cube
zbMATH Keywords: compact Hausdorff spaces
zbMATH Keywords: Hölder's theorem
zbMATH Keywords: Chang's completeness theorem
zbMATH Keywords: Wóicicki's theorem
zbMATH Keywords: rational polyhedra
zbMATH Keywords: piecewise linear maps
zbMATH Keywords: \({\mathbb{Z}}\)-maps
MaRDI profile type: Publication
full work available at URL: s11225-012-9377-z
OpenAlex ID: W1991966428
cites work: Groupes et anneaux reticules
cites work: Q4210476
cites work: Q3934450
cites work: Algebraic Analysis of Many Valued Logics
cites work: A New Proof of the Completeness of the Lukasiewicz Axioms
cites work: Q3994935
cites work: Q4316881
cites work: Q4049062
cites work: The Lebesgue state of a unital abelian lattice-ordered group
cites work: Duality, projectivity, and unification in Łukasiewicz logic and MV-algebras
cites work: Advanced Łukasiewicz calculus and MV-algebras
cites work: Fragments of Many-Valued Statement Calculi
cites work: Q3935151
cites work: The Theory of Representation for Boolean Algebras
cites work: The future of mathematics
DOI: 10.1007/S11225-012-9377-Z
description: scientific article; zbMATH DE number 6088860