Projective MV-algebras and rational polyhedra
type: Item
instance of: scholarly article
title: Projective MV-algebras and rational polyhedra
zbMATH Open document ID: 1196.06004
author: Leonardo Manuel Cabrer
author: Daniele Mundici
published in: Algebra Universalis
publication date: 2010-06-16T00:00:00Z
review text: This paper deals with the geometry of (always finitely generated) projective MV-algebras. The free \(n\)-generator MV-algebra \(\mathcal{M}([0,1]^{n})\) of McNaughton functions on the \(n\)-cube is a trivial example of a finitely generated projective MV-algebra. A rational polyhedron \(P\subseteq [0,1]^{n}\) is defined as the pointset union of finitely many simplexes in \([0,1]^{n}\) with rational vertices. The main result of this paper is that if \(M\) is an \(n\)-generator projective MV-algebra, then there is a rational polyhedron \(P\) in the \(n\)-cube \([0,1]^{n}\) such that \(M\) is isomorphic to the MV-algebra \(\mathcal{M}(P)\) of restrictions to \(P\) of the McNaughton functions of the free \(n\)-generator MV-algebra. \(P\) necessarily contains a vertex \(v_{P}\) of the \(n\)-cube. It is well known that we can obtain from this that every finitely generated projective MV-algebra is finitely presented. The converse does not hold: for instance, if the rational polyhedron \(P\) does not contain any point of denominator 1, or \(P\) is not connected, then the finitely presented MV-algebra \(\mathcal{M}(P)\) is not projective. In this paper the authors characterize those polyhedra contained in the \(n\)-cube such that \(\mathcal{M}(P)\) is projective. In particular, if the rational polyhedron \(P\) is a union of segments originating at some fixed vertex \(v_{P}\) of the \(n\)-cube, then \(\mathcal{M}(P)\) is projective. Using this result, in the final section of the paper, the authors introduce two geometric constructions of new projective MV-algebras from old ones. They prove that if \(A=\mathcal{M}(P)\) and \(B=\mathcal{M}(Q)\) are projective, then so is the subalgebra of \(A\times B\) given by \(\{(f,g)\mid f(v_{P})=g(v_{Q})\}\), and so is the free product \(A\sqcup B\). The authors end their paper with the remark that in view of the categorical equivalence between MV-algebras and unital lattice-ordered abelian groups, the results of this paper have an equivale…
reviewed by: Florentina Chirteş
Mathematics Subject Classification ID: 06D35
Mathematics Subject Classification ID: 06F20
Mathematics Subject Classification ID: 08B30
Mathematics Subject Classification ID: 57Q05
Mathematics Subject Classification ID: 57R05
zbMATH DE Number: 5721487
zbMATH Keywords: MV-algebra
zbMATH Keywords: projective MV-algebra
zbMATH Keywords: rational polyhedron
zbMATH Keywords: retraction
zbMATH Keywords: star-shaped polyhedron
zbMATH Keywords: unimodular triangulation
MaRDI profile type: Publication
full work available at URL: s00012-010-0039-6
OpenAlex ID: W1964644649
cites work: Free Vector Lattices
cites work: Applications of Duality in the Theory of Finitely Generated Lattice-Ordered Abelian Groups
cites work: Q5571891
cites work: Algebraic foundations of many-valued reasoning
cites work: Q5614468
cites work: Q4200260
cites work: Interpretation of AF \(C^*\)-algebras in Łukasiewicz sentential calculus
cites work: Farey stellar subdivisions, ultrasimplicial groups, and \(K_ 0\) of AF \(C^*\)-algebras
cites work: Free products in the category of Abelian \(\ell\)-groups with strong unit
cites work: Q5667781
DOI: 10.1007/S00012-010-0039-6
description: scientific article; zbMATH DE number 5721487