## 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

### Citations:

Zbl 0937.06009; Zbl 0385.03051; Zbl 0937.03030
Full Text:

### References:

 [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
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