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.


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


[1] Chajda, I., Halaš, R., Kühr, J.: Distributive lattices with sectionally antitone involutions. Acta Sci. Math. (Szeged) 71, 19–33 (2005) · Zbl 1099.06006
[2] Chang, C.C.: Algebraic analysis of many-valued logics. Trans. Am. Math. Soc. 88, 467–490 (1958) · Zbl 0084.00704
[3] Cignoli, R.L.O., D’Ottaviano, I.M.L., Mundici, D.: Algebraic Foundations of Many-Valued Reasoning. Kluwer, Dordrecht (2000) · Zbl 0937.06009
[4] Imai, Y., Iséki, K.: On axiom system of propositional calculi. Proc. Jpn. Acad. 42, 19–22 (1966) · Zbl 0156.24812
[5] Iséki, K., Tanaka, S.: An introduction to theory of BCK-algebras. Math. Jpn. 28, 1–26 (1978) · Zbl 0385.03051
[6] Komori, Y.: The class of BCC-algebras is not a variety. Math. Jpn. 29, 391–394 (1984) · Zbl 0553.03046
[7] Mundici, D.: MV-algebras are categorically equivalent to bounded commutative BCK-algebras. Math. Jpn. 31, 889–894 (1986) · Zbl 0633.03066
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.