The writing of the MV-algebras
type: Item
instance of: scholarly article
title: The writing of the MV-algebras
zbMATH Open document ID: 0966.06014
DOI: 10.1023/A:1005069911813
author: C. C. Chang
published in: Studia Logica
publication date: 2001-07-26T00:00:00Z
review text: Chang invented MV-algebras to give a neat, algebraic proof of the completeness of the Łukasiewicz axioms, after the syntactic proof of \textit{A. Rose} and \textit{J. B. Rosser} [Trans. Am. Math. Soc. 87, 1-53 (1958; Zbl 0085.24303)]. His completeness theorem shows that an equation holds for all MV-algebras iff it holds for the single MV-algebra given by the unit real interval \([0,1]\) equipped with negation \(1-x\) and truncated addition. He achieved this in two steps [as witnessed by his papers in Trans. Am. Math. Soc. 88, 467-490 (1958; Zbl 0084.00704) and Trans. Am. Math. Soc. 93, 74-80 (1959; Zbl 0093.01104)]. In the short paper under review, written in a friendly and non-technical style, Chang recalls the origins of MV-algebras. While shifting from Cornell to Los Angeles, and from snow and ice to milder climate, Chang was also shifting from Polish notation to equations, and from maximal to prime ideals. After several years of relative tranquillity, starting from the mid-eighties MV-algebras have become again the object of intense study: since the year 2000 the MSC Classification index has the new special entry 06D35 for them. This is so because MV-algebras are deeply related to lattice-ordered abelian groups (for his proof, Chang had already constructed a functor from MV-chains to totally ordered abelian groups with a distinguished positive element), AF \(C^*\)-algebras, toric varieties and adaptive error-correcting codes [see, e.g., the monograph: \textit{R. L. O. Cignoli, I. M. L. D'Ottaviano} and \textit{D. Mundici}, Algebraic foundations of many-valued reasoning, Kluwer, Dordrecht, Trends in Logic, Studia Logica Library, Vol. 7 (2000; Zbl 0937.06009), as well as \textit{P. Hajek}'s book, Metamathematics of fuzzy logic, Kluwer, Dordrecht, Trends in Logic, Studia Logica Library, Vol. 4 (1998; Zbl 0937.03030)].
reviewed by: Daniele Mundici
Mathematics Subject Classification ID: 06D35
Mathematics Subject Classification ID: 01A60
Mathematics Subject Classification ID: 03G25
Mathematics Subject Classification ID: 03-03
Mathematics Subject Classification ID: 06-03
Mathematics Subject Classification ID: 03B50
zbMATH DE Number: 1512009
zbMATH Keywords: Łukasiewicz logic
zbMATH Keywords: MV-algebras
MaRDI profile type: Publication
full work available at URL: a:1005069911813
OpenAlex ID: W1493087039
description: scientific article; zbMATH DE number 1512009