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Adding structure to MV-algebras. (English) Zbl 0992.06012

Building on the first author’s paper “An algebraic approach to propositional fuzzy logic” [J. Logic Lang. Inf. 9, 91-124 (2000; Zbl 0942.06006)], the authors of this interesting paper consider various enrichments of MV-algebras. These enriched structures are intended to yield an algebraic semantics for extensions of Łukasiewicz infinite-valued logic, where one also incorporates extra t-norms and their residues. For background on the logic of MV-algebras and of t-norms, respectively see the monographs: R. L. O. Cignoli, I. M. L. D’Ottaviano and D. Mundici, Algebraic foundations of many-valued reasoning [Trends in Logic, Studia Logica Library. 7. Dordrecht: Kluwer Academic Publishers (2000; Zbl 0937.06009)], and P. Hajek, Metamathematics of fuzzy logic [Trends in Logic, Studia Logica Library. 4. Dordrecht: Kluwer Academic Publishers (1998; Zbl 0937.03030)]. As proved by the authors of the present paper, in many cases the forgetful functor has a left adjoint, and hence for every MV-algebra \(A\) there is a freest enriched algebra which is generated by \(A\) under the added operations and constraining equations. A major obstruction is the unsolved problem, going back to Birkhoff and Pierce, of characterizing free algebras in the variety generated by the unit real interval \([0,1]\) equipped with negation \(1-x\), truncated sum, and multiplication.

MSC:

06D35 MV-algebras
03B50 Many-valued logic
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