The writing of the MV-algebras
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)].
scientific article; zbMATH DE number 1512009