The going up and going down theorems in MV-algebras and abelian \(l\)-groups
type: Item
instance of: scholarly article
title: The going up and going down theorems in MV-algebras and abelian \(l\)-groups
zbMATH Open document ID: 0949.06003
author: Peter L. Belluce
published in: Journal of Mathematical Analysis and Applications
publication date: 2000-06-04T00:00:00Z
review text: The well-known going up and going down theorems in commutative ring theory relate the prime ideals of a subring to the prime ideals of the overring. The author shows that the going up theorem holds in all MV-algebras, while this is not true in general for the going down theorem. MV-algebras are the Lindenbaum algebras of infinite-valued Łukasiewicz logic, and are deeply related to AF \(C^*\)-algebras, i.e., limits of finite-dimensional \(C^*\)-algebras. For a comprehensive account see the monograph of \textit{R. Cignoli, I. M. L. D'Ottaviano} and \textit{D. Mundici} [Algebraic foundations of many-valued reasoning (Trends in Logic -- Studia Logica Library 7, Kluwer, Dordrecht) (2000; Zbl 0937.06009)]. The author also shows that the lying over theorem [see \textit{I. Kaplansky}, Commutative rings. 2nd ed. (Univ. Chicago Press, Chicago) (1974; Zbl 0296.13001)] holds in all MV-algebras. Using the present reviewer's categorical equivalence between MV-algebras and abelian lattice-ordered groups with strong units [\textit{D. Mundici}, J. Funct. Anal. 65, 15-63 (1986; Zbl 0597.46059)], the author extends his results to this latter class of structures. His main tool is an appropriate notion of ``localization'' at a prime ideal.
reviewed by: Daniele Mundici
Mathematics Subject Classification ID: 06D35
Mathematics Subject Classification ID: 06F20
Mathematics Subject Classification ID: 13B24
Mathematics Subject Classification ID: 13A15
zbMATH DE Number: 1423872
zbMATH Keywords: Gamma functor
zbMATH Keywords: localization
zbMATH Keywords: going down theorem
zbMATH Keywords: going up theorem
zbMATH Keywords: MV-algebras
zbMATH Keywords: lying over theorem
zbMATH Keywords: abelian lattice-ordered groups
zbMATH Keywords: prime ideal
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6623
OpenAlex ID: W2011685121
cites work: Semi-simple and complete MV-algebras
cites work: Yosida Type Representation for Perfect MV‐Algebras
cites work: The Prime Spectrum of an MV‐Algebra
cites work: Groupes et anneaux reticules
cites work: Algebraic Analysis of Many Valued Logics
cites work: Boolean products of MV-algebras: hypernormal MV-algebras
cites work: The poset of prime \(l\)-ideals of an abelian \(l\)-group with a strong unit
cites work: Perfect MV-algebras are categorically equivalent to Abelian \(l\)-groups
cites work: Q4836216
cites work: Q4107488
cites work: Q5603302
cites work: Q4868603
cites work: Interpretation of AF \(C^*\)-algebras in Łukasiewicz sentential calculus
DOI: 10.1006/JMAA.1999.6623
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description: scientific article; zbMATH DE number 1423872