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A basic algebra is an MV-algebra if and only if it is a BCC-algebra. (English) Zbl 1145.06003

Basic algebras are the algebras of P. Hájek’s Basic Logic (see his monograph [Metamathematics of fuzzy logic. Dordrecht: Kluwer (1998; Zbl 0937.03030)]). MV-algebras were introduced by C. C. Chang to give a proof of the completeness theorem for Łukasiewicz infinite-valued propositional logic (see the monograph [R. L. O. Cignoli, I. M. L. D’Ottaviano and D. Mundici, Algebraic foundations of many-valued reasoning. Dordrecht: Kluwer (2000; Zbl 0937.06009)]). BCK-algebras were introduced by K. Iséki and S. Tanaka [Math. Jap. 23, 1–26 (1978; Zbl 0385.03051)]. BCC-algebras are BCK-algebras satisfying the Exchange Identity: \(x\to (y\to z)= y\to (x\to z)\).
In the paper under review the authors prove that a basic algebra is an MV-algebra iff it is a BCC-algebra.

MSC:

06D35 MV-algebras
06F35 BCK-algebras, BCI-algebras
03G25 Other algebras related to logic
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References:

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