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**Implication in MV-algebras.**
*(English)*
Zbl 1097.06011

The class of implication algebras (also known as Tarski algebras) introduced by Abbott form the algebraic counterpart of the implicational fragment of classical propositional logic, i.e., the implication algebras are \(\{\to\}\)-subreducts of Boolean algebras. The authors of this paper generalize the notion of implication algebra in order to obtain similar algebras for Łukasiewicz logic. For this purpose they define a weak implication algebra which arises from the definition of implication algebra by replacing the defining identity \((x\to y)\to x=x\) by identities \(x\to 1=1\) and \(1\to x=x\). Then they characterize the class of weak implication algebras as a certain class of join-semilattices where each interval of the form \([p,1]\) is an MV-algebra. Some of the presented results partially overlap with some known results from the theory of BCK-algebras.

In fact, the class of weak implication algebras as defined in this paper coincides with the class of commutative BCK-algebras [see, e.g., K. Iseki and S. Tanaka, “An introduction to the theory of BCK-algebras”, Math. Jap. 23, 1–26 (1978; Zbl 0385.03051)]. Another paper closely related to these results is [D. Mundici, “MV-algebras are categorically equivalent to bounded commutative BCK-algebras”, Math. Jap. 31, 889–894 (1986; Zbl 0633.03066)], where it is proved that each bounded commutative BCK-algebra is an MV-algebra. It should also be mentioned that the class of commutative BCK-algebras (= weak implication algebras) does not correspond to the implicational fragment of Łukasiewicz logic (i.e., they are not \(\{\to\}\)-subreducts of MV-algebras). It is known that in order to axiomatize such a class we have to add the prelinearity axiom. The fact that the prelinearity axiom is necessary was shown by Y. Komori [“The separation theorem of the \(\aleph_0\)-valued Łukasiewicz propositional logic”, Rep. Fac. Sci., Shizuoka Univ. 12, 1–5 (1978; Zbl 0377.02021)].

In fact, the class of weak implication algebras as defined in this paper coincides with the class of commutative BCK-algebras [see, e.g., K. Iseki and S. Tanaka, “An introduction to the theory of BCK-algebras”, Math. Jap. 23, 1–26 (1978; Zbl 0385.03051)]. Another paper closely related to these results is [D. Mundici, “MV-algebras are categorically equivalent to bounded commutative BCK-algebras”, Math. Jap. 31, 889–894 (1986; Zbl 0633.03066)], where it is proved that each bounded commutative BCK-algebra is an MV-algebra. It should also be mentioned that the class of commutative BCK-algebras (= weak implication algebras) does not correspond to the implicational fragment of Łukasiewicz logic (i.e., they are not \(\{\to\}\)-subreducts of MV-algebras). It is known that in order to axiomatize such a class we have to add the prelinearity axiom. The fact that the prelinearity axiom is necessary was shown by Y. Komori [“The separation theorem of the \(\aleph_0\)-valued Łukasiewicz propositional logic”, Rep. Fac. Sci., Shizuoka Univ. 12, 1–5 (1978; Zbl 0377.02021)].

Reviewer: Rostislav Horcik (Praha)