ResumoIn  it is proved the categorical isomorphism of two varieties: bounded commutative BCK-algebras and MV -algebras. The class of MV -algebras is the algebraic counterpart of the infinite valued propositional calculus L of Lukasiewicz (see ). The main objective of the present paper is to study that isomorphism from the perspective of logic. The B-C-K logic is algebraizable and the quasivariety of BCKalgebras is the equivalent algebraic semantics for that logic (see ). We call commutative B-C-K logic, briefly cBCK, to the extension of B-C-K logic associated to the variety of commutative BCK–algebras. Moreover, we present the extension Boc of cBCK obtained by adding the axiom of “boundness”. We prove that the deductive system Boc is equivalent to L. We observe that cBCK admits two interesting extensions: the logic Boc, treated in this paper, which is equivalent to the system L of Lukasiewicz, and the logic Co that is naturally associated to the system Balo of `-groups (see , ) . This constructions establish a link between L and Balo , that would be a logical approach to the categorical relationship between MV–algebras and `-groups (see ).
BLOK, W.J., PIGOZZI, D. “Algebraizable Logics”. Memoirs of the A.M.S., 77, Nr. 396, 1989.
CHANG, C.C. “Algebraic Analysis of Many-Valued Logics”. Transactions of the A.M.S., 88, pp. 467-490, 1958.
———–. “A New Proof of the Completeness of the Lukasiewicz Axioms”. Transactions of the A.M.S., 93, pp. 74-80, 1959.
CIGNOLI, R., D’OTTAVIANO, I.M.L., MUNDICI, D. Algebraic Foundations of many-valued Reasoning. Trends in Logic, Studia Logic Library, vol. 7, Kluwer Ac. Publ., 2000.
GALLI, A., LEVIN, R., SAGASTUME, M. “The Logic of Equilibrium and Abelian Lattice Ordered Groups”. Arch. Math Logic, 43, pp. 141-158, 2004.
ISEKI, K. “An Algebra Related with a Propositional Calculus”. Proc. Japan Acad., 42, pp. 26-29, 1966.
ISEKI, K., TANAKA, S. “An Introduction to the Theory of CKalgebras”.
Math. Japon, 23, pp. 1-26, 1978.
MANGANI, P. “Su Certe Algebre Connesse con Logiche a Pi´u Valori”. Bollettino Unione Matematica Italiana, 8, pp. 68-78, 1973.
MUNDICI, D. “MV-algebras are Categorically Equivalent to Bounded Commutative BCK-algebras”. Math. Japonica, 31(6), pp. 889-894, 1986.
SAGASTUME, M., “Conical Logic and l-groups Logic”. JANCL, 15(3), pp. 265-283, 2005.
WRONSKI, A. “BCK-algebras do Not Form a Variety”. Math. Japonica, 28, pp. 211-213, 1983.
YUTANI, H. “Quasi-commutative BCK-algebras and Congruence
Relations”. Math. Seminar Notes, 5, pp. 469-480, 1977.